Ir source and phased antenna with graphene layer and related methods

ABSTRACT

An IR source may also include an electrically conductive layer defining a back contact. The IR source may also include a first dielectric layer over the electrically conductive layer, a transparent electrically conductive layer over the first dielectric layer, and a second dielectric layer over the transparent electrically conductive layer. The IR source may include a graphene layer over the second dielectric layer and having a perforated pattern, a protective layer over the graphene layer, and first and second electrically conductive contacts coupled to the graphene layer. The graphene layer may be configured to emit IR radiation when a voltage signal is applied between the first and second electrically conductive contacts.

RELATED APPLICATION

This application is based upon prior filed copending Application No.63/059,456 filed Jul. 31, 2020, the entire subject matter of which isincorporated herein by reference in its entirety.

GOVERNMENT RIGHTS

This invention was made with government support under contract numberNSF-CISE-1514089 awarded by National Science Foundation. The governmenthas certain rights in the invention.

TECHNICAL FIELD

The present disclosure relates to the field of radiation sources, and,more particularly, to an infrared source and related methods.

BACKGROUND

Infrared (IR) sources are useful in many applications. For example, IRsources can be used in spectroscopy applications, and used to irradiatean unknown object. In some applications, IR sources are used forcommunications in electro-optic platforms. In some recent advances,near-IR (NIR) spectroscopy has been used to detect viral matter.

In some approaches, graphene has been used in IR applications. Graphene,one of the widely studied two dimensional materials, comprises a singlelayer of carbon atoms in a honeycomb lattice. It has special electrical,optical, and mechanical properties due to its tunable band dispersionrelation and atomic thickness. Because of its unique band structure,graphene possesses very high mobility and fast carrier relaxation time,making it an attractive candidate for ultrafast electronics andoptoelectronic devices such as transistors, optical switches,mid-infrared (MIR) photodetectors, photovoltaic devices, saturatedabsorbers and ultrafast lasers, etc.

For example, U.S. Pat. Nos. 10,784,387 and 10,312,389 are assigned tothe present application's assignee and each discloses an opticaldetector device including a substrate, and a reflector layer carried bythe substrate. The optical detector device comprises a first dielectriclayer over the reflector layer, and a graphene layer over the firstdielectric layer and having a perforated pattern therein.

SUMMARY

Generally, an IR source may include an electrically conductive layer, afirst dielectric layer over the electrically conductive layer, atransparent electrically conductive layer over the first dielectriclayer, a second dielectric layer over the transparent electricallyconductive layer, and a graphene layer over the second dielectric layerand having a perforated pattern. The IR source may further comprisefirst and second electrically conductive contacts coupled to thegraphene layer. The graphene layer may be configured to emit IRradiation in a frequency range when a voltage signal is applied betweenthe first and second electrically conductive contacts.

More specifically, the graphene layer may be configured to emit the IRradiation at an angle from normal based upon the voltage signal. Thegraphene layer may be configured to emit the IR radiation at the anglewithin a range of 12°-80° from normal based upon the voltage signal. Thegraphene layer may be configured to selectively change the frequencyrange based upon the voltage signal.

In some embodiments, the perforated pattern may comprise an array ofelliptical holes. The graphene layer may be configured to emit mid-IRradiation. For example, the electrically conductive layer may compriseat least one of gold, silver, and platinum, and the first dielectriclayer may comprise a polymer layer, and wherein the second dielectriclayer comprises an oxide layer. The electrically conductive layer may becoupled to a reference voltage, for example, a ground potential.

Another aspect is directed to a phased array. The phased array mayinclude an electrically conductive layer, a first dielectric layer overthe electrically conductive layer, and a transparent electricallyconductive layer over the first dielectric layer. The phased array mayalso comprise a second dielectric layer over the transparentelectrically conductive layer, and a graphene layer over the seconddielectric layer and having a perforated pattern comprising an array ofelliptical holes. The graphene layer may be configured to emit IRradiation in a frequency range. The phased array also may comprise firstand second electrically conductive contacts coupled to the graphenelayer, and circuity. The circuity may be configured to apply a voltagesignal between the first and second electrically conductive contacts,change the voltage signal to selectively set phase change between the IRradiation emitted from adjacent elliptical holes to emit the IRradiation at an angle from normal, and change the voltage signal toselectively set the frequency range of the IR radiation.

Another aspect is directed to a method for making an IR source. Themethod may comprise forming a first dielectric layer over anelectrically conductive layer, forming a transparent electricallyconductive layer over the first dielectric layer, and forming a seconddielectric layer over the transparent electrically conductive layer. Themethod may further include forming a graphene layer over the seconddielectric layer and having a perforated pattern, and forming first andsecond electrically conductive contacts coupled to the graphene layer.The graphene layer may be configured to emit IR radiation in a frequencyrange when a voltage signal is applied between the first and secondelectrically conductive contacts.

Yet another aspect is directed to a method of operating a phased arraycomprising an electrically conductive layer, a first dielectric layerover the electrically conductive layer, a transparent electricallyconductive layer over the first dielectric layer, a second dielectriclayer over the transparent electrically conductive layer, a graphenelayer over the second dielectric layer and having a perforated patterncomprising an array of elliptical holes, the graphene layer configuredto emit IR radiation in a frequency range, and first and secondelectrically conductive contacts coupled to the graphene layer. Themethod may include operating circuity coupled between the first andsecond electrically conductive contacts to apply a voltage signalbetween the first and second electrically conductive contacts, changethe voltage signal to selectively set phase change between the IRradiation emitted from adjacent elliptical holes to emit the IRradiation at an angle from normal, and change the voltage signal toselectively set the frequency range of the IR radiation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an image of an ultrafast mid-IR light source based onpatterned graphene placed on top of a cavity, which can be tuned bymeans of a gate voltage applied to the indium tin oxide layer, accordingto the present disclosure.

FIG. 2 is a schematic diagram of an example embodiment of the IR source,according to the present disclosure.

FIG. 3 is a diagram of emittance in the example embodiment of the IRsource, according to the present disclosure.

FIG. 4 is a diagram of emittance in the example embodiment of the IRsource, according to the present disclosure.

FIG. 5 is a diagram of emittance in the example embodiment of the IRsource, according to the present disclosure.

FIG. 6 is a diagram of spectral radiance in the example embodiment ofthe IR source, according to the present disclosure.

FIG. 7 is a diagram of spectral radiance in the example embodiment ofthe IR source, according to the present disclosure.

FIG. 8A is a diagram of temperature distribution in the exampleembodiment of the IR source, according to the present disclosure.

FIG. 8B is as spatial diagram of temperature distribution in the exampleembodiment of the IR source, according to the present disclosure.

FIG. 9 is a diagram of spectral radiance in the example embodiment ofthe IR source, according to the present disclosure.

FIG. 10 is a spherical density plot in the example embodiment of the IRsource, according to the present disclosure.

FIG. 11 is a diagram of coherence length in the example embodiment ofthe IR source, according to the present disclosure.

FIGS. 12A-12D are diagrams of directivity in the example embodiment ofthe IR source, according to the present disclosure.

FIG. 13 is a diagram of theoretical fit to spectral radiance in theexample embodiment of the IR source, according to the presentdisclosure.

FIG. 14 is a diagram of spectral reflectance in the example embodimentof the IR source, according to the present disclosure.

FIG. 15 is a diagram of spectral reflectance in the example embodimentof the IR source, according to the present disclosure.

FIG. 16 is a diagram of spectral reflectance in the example embodimentof the IR source, according to the present disclosure.

FIG. 17 is a diagram of a single-sheet hyperboloid with an ellipticalwormhole, according to the present disclosure.

FIG. 18 is a diagram of energy loss function in the example embodimentof the IR source, according to the present disclosure.

FIG. 19A is a schematic side view of an example embodiment of the IRsource, according to the present disclosure.

FIG. 19B is a schematic top plan view of the perforated pattern and thefirst and second electrically conductive contacts of the IR source ofFIG. 19A.

FIG. 20 is a schematic side view of an example embodiment of a phaseantenna array, according to the present disclosure.

FIG. 21 is a flowchart of a method for making the IR source of FIG. 19A.

FIG. 22 is a flowchart of a method of operating the phased array of FIG.20.

DETAILED DESCRIPTION

The present disclosure will now be described more fully hereinafter withreference to the accompanying drawings, in which several embodiments ofthe invention are shown. This present disclosure may, however, beembodied in many different forms and should not be construed as limitedto the embodiments set forth herein. Rather, these embodiments areprovided so that this disclosure will be thorough and complete, and willfully convey the scope of the present disclosure to those skilled in theart. Like numbers refer to like elements throughout, and base 100reference numerals are used to indicate similar elements in alternativeembodiments.

An object that is kept in equilibrium at a given temperature T>0 K emitselectromagnetic (EM) radiation because the charge carriers on the atomicand molecular scale oscillate due to their heat energy. [3] Planck's lawdescribes quantitatively the energy density u(ω) of the EM radiation perunit frequency ω for black-body radiation, which is

${{u_{BB}(\omega)}d\omega} = \frac{\omega^{2}}{\pi^{2}c^{3}}$

Θ(ω)dω, where c is the speed of light in vacuum, ℏ is the Planckconstant, and k_(B) is the Boltzmann constant.Θ(ω,T)=ℏω/[exp(ℏω/k_(B)T)−1] is the thermal energy of a photon mode.Consequently, the energy emitted per unit surface area and per unitfrequency, also called spectral radiance, of a black body intothree-dimensional (3D) space is given by

$\begin{matrix}{{{I_{BB}(\omega)}d\omega} = {{\frac{1}{4\pi}c{u(\omega)}} = {\frac{\omega^{2}}{4\pi^{3}c^{2}}{\Theta(\omega)}d\;{\omega.}}}} & (1)\end{matrix}$

The total energy density u can then be obtained by integrating over allfrequencies and angles over the half-sphere, leading to theStefan-Boltzmann law for the energy density of black-body radiation,

$\begin{matrix}{{u_{BB} = {{\left( \frac{8\pi^{5}k_{B}^{4}}{15c^{3}h^{3}} \right)T^{4}} = {a_{BB}T^{4}}}},} & (2)\end{matrix}$

with a_(BB)=7.566×10⁻¹⁶ Jm⁻³K⁻⁴. The total power emitted per unitsurface area P/A of a black-body is

$\begin{matrix}\begin{matrix}{I_{BB} = {\frac{P}{A} = {\int\limits_{0}^{\infty}{{I_{BB}(\omega)}d\;\omega{\int\limits_{0}^{2\pi}{d\;\varphi{\int\limits_{0}^{\pi\text{/}2}{\cos\;\theta\;\sin\;\theta\; d\;\theta}}}}}}}} \\{= {{\pi{\int\limits_{0}^{\infty}{{I_{BB}(\omega)}d\;\omega}}} = {\frac{1}{4\pi}{uc}}}} \\{{= {{\frac{a_{BB}c}{4\pi}T^{4}} = {{b_{BB}T^{4}} = {\left( \frac{\pi^{2}k_{B}^{4}}{60c^{2}\hslash^{3}} \right)T^{4}}}}},}\end{matrix} & (3)\end{matrix}$

where b_(BB)=5.67×10⁻⁸ Wm⁻²K⁻⁴ is the Stefan-Boltzmann constant. Thefactor cos θ is due to the fact that black bodies are Lambertianradiators.

In recent years, several methods have been implemented for achieving aspectrally selective emittance, in particular narrowband emittance,which increases the coherence of the emitted photons. One possibility isto use a material that exhibits optical resonances due to the bandstructure or due to confinement of the charge carriers. [3] Anothermethod is to use structural optical resonances to enhance and/orsuppress the emittance. Recently, photonic crystal structures have beenused to implement passive pass band filters that reflect the thermalemission at wavelengths that match the photonic bandgap. [8, 17]Alternatively, a truncated photonic crystal can be used to enhance theemittance at resonant frequencies. [6, 28]

Recent experiments have shown that it is possible to generate infrared(IR) emission by means of Joule heating created by means of a biasvoltage applied to graphene on a SiO₂/Si substrate. [9, 20] In order toavoid the breakdown of the graphene sheet at around T=700 K, thegraphene sheet can be encapsulated between hexagonal boron nitride(h-BN) layers, which remove efficiently the heat from graphene. The toplayer protects it from oxidation. [14, 19] In this way, the graphenesheet can be heated up to T=1600 K, [19] or even above T=2000 K. [14,26] Kim et al. and Luo et al. [30] demonstrated broadband visibleemission peaked around a wavelength of λ=725 nm. [14, 19] By using aphotonic crystal substrate made of Si, Shiue et al. [26] demonstratednarrowband near-IR emission peaked at around λ=1600 nm with an emittanceof around ϵ=0.07. [26] To the best of our knowledge, there are neithertheoretical nor experimental studies on spectrally selective thermalemission from graphene in the mid-IR range.

Here, we present the proof of concept of a method to tune the spectrallyselective thermal emission from nanopatterned graphene (NPG) by means ofa gate voltage that varies the resonance wavelength of localized surfaceplasmons (LSPs) around the circular holes that are arranged in ahexagonal or square lattice pattern in a single graphene sheet in thewavelength regime between 3 μm and 12 μm. By generalizing Planck'sradiation theory to grey-body emission, we show that the thermalemission spectrum can be tuned in or out of the two main atmospherictransparency windows of 3 to 5 μm and 8 to 12 μm in the mid-IR regime,and also in or out of the opaque mid-IR regime between 5 and 8 μm. Inaddition, the gate voltage can be used to tune the direction of thethermal emission due to the coherence between the localized surfaceplasmons (LSPs) around the holes due to the nonlocal response functionin graphene, which we show by means of a nonlocalfluctuation-dissipation theorem. The main element of the nanostructureis a circular hole of diameter a in a graphene sheet. Therefore let usfocus first on the optoelectronic properties of a single hole.

The frequency-dependent dipole moment of the hole is

$\begin{matrix}\begin{matrix}{{p\left( {r,\omega} \right)} = {{- ɛ_{0}}{ɛ\left( {r,\omega} \right)}E_{0{}}}} \\{{= {{- {\alpha_{1,2}\left( {r,\omega} \right)}}E_{0{}}}},}\end{matrix} & (4)\end{matrix}$

where the polarizabilities α_(1,2) are given along the main axes x and yof the elliptic hole, and r=r₀ is the position of the dipole moment,i.e. the hole. Graphene's dielectric function is isotropic in thexy-plane, i.e. ε″_(∥)=ε″_(xx)=ε″_(yy). V₀ is the volume of the graphenesheet. In the Supplementary Information we derive the generalpolarizabilities of an uncharged single-sheet hyperboloid withdielectric function ε(ω) inside a medium with dielectric constant ε_(m)[see Eq. (163)]. The polarizabilities of an elliptical wormhole in x-and y-direction read

$\begin{matrix}{{{\alpha_{1}(\omega)} = {\frac{2{abd}\;{\pi\left( {{\pi\text{/}2} - 1} \right)}}{3}\frac{{ɛ(\omega)} - ɛ_{m}}{ɛ_{m} + {L_{1}\left\lbrack {{ɛ(\omega)} - ɛ_{m}} \right\rbrack}}}},} & (5) \\{{\alpha_{2}(\omega)} = {\frac{2{abd}\;{\pi\left( {{\pi\text{/}2} - 1} \right)}}{3}{\frac{{ɛ(\omega)} - ɛ_{m}}{ɛ_{m} + {L_{2}\left\lbrack {{ɛ(\omega)} - ɛ_{m}} \right\rbrack}}.}}} & (6)\end{matrix}$

respectively, for which the in-plane polarizabilities lies in the planeof the graphene sheet that is parallel to the xy-plane. ε(ω) is thedielectric function of graphene. We assumed that the thickness d of thegraphene sheet is much smaller than the size of the elliptic hole. Thegeometrical factors in this limit are

$\begin{matrix}{{L_{1} \approx {{abd}{\int_{\eta_{1}}^{\infty}\frac{d\;\eta^{\prime}}{\left( {\eta^{\prime} + a^{2}} \right)R_{\eta^{\prime}}}}}},} & (7) \\{L_{2} \approx {{abd}{\int_{\eta_{1}}^{\infty}{\frac{d\;\eta^{\prime}}{\left( {\eta^{\prime} + b^{2}} \right)R_{\eta^{\prime}}}.}}}} & (8)\end{matrix}$

In the case of a circular hole of diameter a the polarizabilitysimplifies to

$\begin{matrix}{{{\alpha_{}(\omega)} = {\frac{2a^{2}d\;{\pi\left( {{\pi\text{/}2} - 1} \right)}}{3}\frac{{ɛ(\omega)} - ɛ_{m}}{ɛ_{m} + {L_{}\left\lbrack {{ɛ(\omega)} - ɛ_{m}} \right\rbrack}}}},} & (9)\end{matrix}$

The LSP frequency of the hole can be determined from the equation

ε_(m) +L _(∥)[ε(ω)−ε_(m)]=0,  (10)

the condition for which the denominator of α_(∥) vanishes.

FIG. 2 includes a schematic showing a ultrafast mid-IR light source 500with the materials used in our setup. The materials from top to bottomare: 1 single layer of hexagonal boron nitride (h-BN), for preventingoxidation of graphene at higher temperatures, 1 single layer ofpatterned graphene, 50 nm of Si₃N₄, for large n-doping and gating, 50 nmof ITO, metallic contact for gating, which is also transparent inmid-IR, λ/4n_(SUB) of SU₈, [24] which is transparent in mid-IR, and Auback mirror. n_(SU8)=1.56 is the refractive index of SU₈.

Using the linear dispersion relation, the intraband optical conductivityis [24, 22]

$\begin{matrix}{{{\sigma_{intra}(\omega)} = {\frac{e^{2}\mspace{14mu} 2k_{B}T}{{{\pi\hslash}^{2}\tau^{- 1}} - {i\;\omega}}{\ln\left\lbrack {2{\cosh\left( \frac{ɛ_{F}}{2k_{B}T} \right)}} \right\rbrack}}},} & (11)\end{matrix}$

which in the case of ε_(F)>>k_(B)T is reduced to

$\begin{matrix}{{{\sigma_{intra}(\omega)} = {{\frac{e^{2}}{\pi\hslash}\frac{\epsilon_{F}}{\tau^{- 1} - {i\;\omega}}} = \frac{2ɛ_{m}\omega_{p}^{2}}{{\pi\hslash}^{2}\left( {\tau^{- 1} - {i\;\omega}} \right)}}},} & (12)\end{matrix}$

where τ is determined by impurity scattering and electron-phononinteraction τ⁻¹=τ_(imp) ⁻¹+τ_(e-ph) ⁻¹. Using the mobility μ of the NPGsheet, it can be presented in the form τ⁻¹=ev_(F) ²/(μE_(F)), wherev_(F)=1 0⁶ m/s is the Fermi velocity in graphene. ω_(p)=√{square rootover (e²ϵ_(F)/2ε_(m))} is the bulk graphene plasma frequency.

It is well-known by now that hydrodynamic effects play an important rolein graphene because the Coulomb interaction collision rate is dominant,i.e. τ_(ee) ⁻¹>>τ_(imp) ⁻¹ and τ_(ee) ⁻¹>>τ_(e-ph) ⁻¹, which correspondsto the hydrodynamic regime. τ_(imp) ⁻¹ and τ_(e-ph) ⁻¹ are theelectron-impurity and electron-phonon collision rates. Since for largeabsorbance and emittance, we choose a large Fermi energy, we are in theFermi liquid regime of the graphene sheet. Taking the hydrodynamiccorrection into account, we also consider the hydrodynamically adjustedintraband optical conductivity, [2, 7]

$\begin{matrix}{{{\sigma_{intra}^{HD}(\omega)} = \frac{\sigma_{intra}(\omega)}{1 - {\eta^{2}\frac{k_{}^{2}}{\omega^{2}}}}},} & (13)\end{matrix}$

where

${\eta^{2} = {\beta^{2} + {D^{2}{\omega\left( {\gamma + {i\;\omega}} \right)}}}},{\beta^{2} \approx {\frac{3}{4}v_{F}^{2}}}$

is the intraband pressure velocity, D≈0.4 μm is the diffusion length ingraphene, and γ=τ⁻¹ is the relaxation rate. Interestingly, the opticalconductivity becomes k-dependent and nonlocal. Also, below we willconjecture that the diffusion length D must be frequency-dependent. Theeffect of the hydrodynamic correction on the LSP resonances diagrams505, 510, 515 at around λ=4 μm, 7 μm, and 10 μm is shown in FIGS. 3, 4,and 5, respectively.

Note that since ε=1+χ, where χ is the susceptibility, it is possible toreplace ε″=χ″. Alternatively, using the formula of the polarizabilityα=ε_(0χ) we can write ε″=α″/ε₀. The dielectric function for graphene isgiven by [24, 22]

$\begin{matrix}{{{ɛ_{}(\omega)} = {ɛ_{g} - \frac{i\;{\sigma_{2D}(\omega)}}{ɛ_{0}\omega\; d}}},} & (14)\end{matrix}$

where ϵ_(g)=2.5 is the dielectric constant of graphite and d is thethickness of graphene. Inserting this formula into Eq. (10) gives

$\begin{matrix}{{{ɛ_{m} + {L_{}\left\lbrack {ɛ_{g} - {i\frac{e^{2}}{{\pi\hslash}^{2}}\frac{\epsilon_{F}}{\tau^{- 1}ɛ_{0}\omega\; d}} - \frac{i}{ɛ_{0}d} - ɛ_{m}} \right\rbrack}} = 0},} & (15)\end{matrix}$

Solving for the frequency and using the real part we obtain the LSPfrequency,

$\begin{matrix}{{{{Re}\;\omega_{LSP}} = \frac{2L_{}^{2}ɛ_{m}\omega_{p}^{2}\tau}{{\pi\hslash}^{2}\left\{ {L^{2} + {d^{2}{ɛ_{0}^{2}\left\lbrack {{L_{}\left( {ɛ_{g} - ɛ_{m}} \right)} + ɛ_{m}} \right\rbrack}^{2}}} \right\}}},} & (16)\end{matrix}$

which is linear in the Fermi energy ϵ_(F).

FIG. 3 shows emittance ϵ(λ) [equal to absorbance A(λ)] of the structureshown in FIGS. 1 and 2 with Fermi energy E_(F)=−1.0 eV, mobility μ=3000V/cm² s, hole diameter of a=30 nm, and period of

=45 nm at T=300 K. The solid (black) curve represents the result of FDTDcalculation. The dashed (blue) curve and the solid (black) curve are theemittances ϵ_(g) and ϵ_(FP), calculated by means of Eq. (26) and Eq.(31) for the bare NPG sheet and the whole structure including cavity,respectively. The dotted (green) line exhibits a blue-shift due to thehydrodynamic correction shown in Eq (13) with D(v=30 THz)≈0. Theblue-shifted dashed (magenta) curve and the blue-shifted dot-dashed(cyan) curve are the RPA-corrected LSP peaks due to the Coulombinteraction and the Coulomb interaction including electron-phononinteraction with the optical phonons of graphene, boron nitride, andSi₃N₄. This NPG sheet emits into the atmospheric transparency windowbetween 3 and 5 μm.

Let us now consider the 2D array of circular holes in a graphene sheet.Since the dipole moments p_(j)=δp(R_(j),ω) interact with each other byinducing dipole moments, we need to consider the dressed dipole momentat each site R_(j) as source of the electric field, which is

{tilde over (p)} _(j) =p _(j)+αΣ_(j′≠j)

_(jj) ,{tilde over (p)} _(j′),  (17)

where

_(jj′) is the dipole-dipole interaction tensor. Using Bloch's theoremp_(j)=p₀ exp(ik_(∥)·R_(∥)), the effective dipole moment becomes

{tilde over (p)} ₀ =p ₀ +{tilde over (p)} ₀αΣ_(j′≠j)

_(jj′) e ^(ik) ^(∥) ^(·(R) ^(j) ^(−R) ^(j′) ⁾.  (18)

for each site j, and thus

$\begin{matrix}{{\overset{\sim}{p}}_{0} = {\frac{p_{0}}{1 - {\alpha\;\mathcal{G}}}.}} & (19)\end{matrix}$

The lattice some over the dipole-dipole interaction tensor

=Σ_(j′≠j)

_(jj′)e^(ik) ^(∥) ^(·(R) ^(j) ^(−R) ^(j′) ⁾ can be found in Ref. [27],i.e.

Re

≈g/

,  (20)

Im

=S−2k ³/3,  (21)

where

is the lattice period,

$\begin{matrix}{S = {\frac{2\pi\; k}{\Omega_{0}} \times \left\{ {\begin{matrix}{\arccos\;\theta} & {{{for}\mspace{14mu} s\mspace{14mu}{polarization}},} \\{{\cos\;\theta}\mspace{31mu}} & {{for}\mspace{14mu} p\mspace{14mu}{{polarization}.}}\end{matrix},} \right.}} & (22)\end{matrix}$

Ω₀ is the unit-cell area, and the real part is valid for periods muchsmaller than the wavelength. The factor g=5.52 (g=4.52) for hexagonal(square) lattice. The electric field created by the effective dipolemoment is determined by

p₀={tilde over (α)}E,  (23)

from which we obtain the effective polarizability of a hole in thecoupled dipole approximation (CDA),

$\begin{matrix}{\overset{\sim}{\alpha} = {\frac{\alpha}{1 - {\alpha\mathcal{G}}}.}} & (24)\end{matrix}$

This formula is the same as in Refs. [31], [27], where the absorption ofelectromagnetic waves by arrays of dipole moments and graphene diskswere considered, respectively. Thus, our result corroborates Kirchhoff'slaw (see below). Consequently, we obtain the same reflection andtransmission amplitudes as in Ref. [27], i.e.

$\begin{matrix}{{r = \frac{{\pm i}S}{\alpha^{- 1} - \mathcal{G}}},{t = {1 + r}},} & (25)\end{matrix}$

where the upper (lower) sign and S=2πω/cΩ₀ cos θ (S=2πΩ cos θ/cΩ₀) applyto s (p) polarization. Thus, the emittance and absorbance of the bareNPG sheet are given by [32] [33]

ϵ_(g) =A _(g)=1−|r| ² −|t| ².  (26)

The coupling to the interface of the substrate with reflection andtransmission amplitudes r₀ and t₀, respectively, which is locatedbasically at the same position as the NPG sheet, yields the combinedreflection and transmission amplitudes [27]

$\begin{matrix}{{R = {r + \frac{{tt}^{\prime}r_{0}}{1 - {r_{0}r^{\prime}}}}},{T = \frac{{tt}_{0}}{1 - {r_{0}r^{\prime}}}},} & (27)\end{matrix}$

where r′=r and t′=1−r are the reflection and transmission amplitudes inbackwards direction, respectively. The results for the LSP resonances ataround λ=4 μm, 7 μm, and 10 μm are shown in FIGS. 2, 3, and 4,respectively.

FIG. 4 shows emittance ϵ(λ) [equal to absorbance A(λ)] of the structureshown in FIGS. 1 and 2 with Fermi energy E_(F)=−1.0 eV, mobility μ=3000V/cm² s, hole diameter of a=90 nm, and period of

=150 nm at T=300 K. The curves are denoted the same as in FIG. 2. ThisNPG sheet emits into the atmospheric opacity window between 5 and 8 μm.

If we include also the whole substrate including cavity and Au mirror,we need to sum over all possible optical paths in the Fabry-Perotcavity, yielding

R _(FP) =R+TT′r _(Au) e ^(iδ)Σ_(m=0) ^(∞) r _(m) ^(m),  (28)

with

r_(m)=r_(Au)R′e^(iδ),  (29)

where r_(Au) is the complex reflection amplitude of the Au mirror in theIR regime. δ=2kL cos θ is the phase accumulated by one back-and-forthscattering inside the Fabry-Perot cavity of length L. k≈n_(SU) ₈ k₀ isthe wavenumber inside the cavity for an external EM wave with wavenumberk₀=2π/λ. Since the sum is taken over a geometric series, we obtain

$\begin{matrix}{{R_{FP} = {R + \frac{{TT}^{\prime}r_{Au}e^{i\;\delta}}{1 - {r_{Au}R^{\prime}e^{i\;\delta}}}}}.} & (30)\end{matrix}$

Since the transmission coefficient through the Au mirror can beneglected, we obtain the emittance ϵ and absorbance A including cavity,i.e.

ϵ_(FP) =A _(FP)=1−|R _(FP)|².  (31)

The results for the LSP resonances at around λ=4 μm, 7 μm, and 10 μm areshown in FIGS. 2, 3, and 4, respectively.

FIG. 5 shows emittance ϵ(λ) [equal to absorbance A(λ)] of the structureshown in FIGS. 1 and 2 with Fermi energy E_(F)=−1.0 eV, mobility μ=3000V/cm² s, hole diameter of a=300 nm, and period of

=450 nm at T=300 K. The curves are denoted the same as in FIG. 2. ThisNPG sheet emits into the atmospheric transparency window between 8 and12 μm.

Using these results, let us consider the excitation of the graphenesheet near the hole by means of thermal fluctuations, which give rise toa fluctuating EM field of a localized surface plasmon (LSP). This can bebest understood by means of the fluctuation-dissipation theorem, whichprovides a relation between the rate of energy dissipation in anon-equilibrium system and the quantum and thermal fluctuationsoccurring spontaneously at different times in an equilibrium system.[21] The standard (local) fluctuation-dissipation theorem forfluctuating currents δĴ_(v)(r,ω) in three dimensions reads

<δĴ _(μ)(r,ω)δĴ _(v)(r′,ω′)>=ωε₀ε″_(μv)(r,ω)Θ(ω)

×δ(ω−ω′)δ(r−r′),  (32)

where the relative permittivity ε(r,ω)=ε′(r,ω)+iε″(r,ω)=f(r)ε(ω) andμ,v=x, y, z are the coordinates. Note that since ε=1+χ, where χ is thesusceptibility, it is possible to replace ε″=χ″. Alternatively, usingthe formula of the polarizability α=ε_(0χ) we can write ε″=α″/ε₀. f(r)=1on the graphene sheet and 0 otherwise. Since the fluctuating currentsare contained inside the two-dimensional graphene sheet, we write thelocal fluctuation-dissipation theorem in its two-dimensional form, i.e.

<δĴ _(μ)(r _(∥),ω)δĴ _(v)(r _(∥)′,ω′)>=σ′_(μv) ^(2D)(r _(∥),ω)Θ(ω)

×δ(ω−ω′)δ(r _(∥) −r _(∥)′),  (33)

where the fluctuating current densities have units of A/m² and thecoordinates are in-plane of the graphene sheet.

Using the method of dyadic Green's functions, it is possible to expressthe fluctuating electric field generated by the fluctuating currentdensity by

δE(r,ω)=iωμ ₀∫_(Ω) G(r,r _(0∥);ω)δJ(r _(0∥),ω)d ² r _(0∥),  (34)

where Ω is the surface of the graphene sheet. The LSP excitation arounda hole can be well approximated by a dipole field such that

$\begin{matrix}{{{\delta\;{J\left( {r_{0||},\omega} \right)}} = {{{- i}\omega{\sum\limits_{j}{\delta{p\left( {R_{j},\omega} \right)}}}} = {{- i}\;{\omega\delta}{p_{0}(\omega)}{\sum_{j}{\delta\left( {r_{0||} - R_{j}} \right)}}}}},} & (35)\end{matrix}$

where R_(j)=(x_(j),y_(j)) are the positions of the holes in the graphenesheet.

FIG. 6 shows spectral radiance diagram 520 of NPG including cavity, asshown in in FIGS. 1 and 2, as a function of wavelength λ with Fermienergy E_(F)=−1.0 eV, mobility μ=3000 V/cm² s, hole diameter of a=30 nm,and period of

=45 nm at 1300 K, 1700 K, and 2000 K.

Consequently, we have

δE(r,ω)=ω²μ₀ δp ₀(ω)Σ_(j) G(r,R _(j);ω).  (36)

The dyadic Green function is defined as

$\begin{matrix}{{\overset{\leftrightarrow}{G}\left( {r,{r^{\prime};\omega}} \right)} = {\left\lbrack {\overset{\leftrightarrow}{11} + {\frac{1}{{k(\omega)}^{2}}{\nabla\nabla}}} \right\rbrack{G\left( {r,{r^{\prime};\omega}} \right)}}} & (37)\end{matrix}$

with the scalar Green function given by

$\begin{matrix}{{{G\left( {r,{r^{\prime};\omega}} \right)} = \frac{e^{{- i}{{k{(\omega)}} \cdot {{r - r^{\prime}}}}}}{4\pi{{r - r^{\prime}}}}},{{{and}\mspace{14mu}{k(\omega)}^{2}} = {{\left( {\omega^{2}/c^{2}} \right)\left\lbrack {{ɛ_{xx}(\omega)},{ɛ_{yy}(\omega)},{ɛ_{ZZ}(\omega)}} \right\rbrack}.}}} & (38)\end{matrix}$

Then, the fluctuation-dissipation theorem can be recast into the forms

$\begin{matrix}{< {\delta{{\overset{\sim}{p}}_{\mu}\left( {r_{0||},\omega} \right)}\delta{{\overset{\sim}{p}}_{\nu}^{*}\left( {r_{0||}^{\prime},\omega^{\prime}} \right)}}>={\frac{\sigma_{\mu\nu}^{\prime 2D}\left( {R_{i},\omega} \right)}{\omega^{2}}{\Theta(\omega)}{\delta\left( {\omega - \omega^{\prime}} \right)} \times {\delta\left( {r_{0||} - r_{0||}^{\prime}} \right)}}} & (39)\end{matrix}$

and thus we obtain

$\begin{matrix}{{< {\delta{{\hat{E}}_{\mu}\left( {r,\omega} \right)}\delta{{\hat{E}}_{\nu}^{*}\left( {r^{\prime},\omega^{\prime}} \right)}}>={\omega^{4}\mu_{0}^{2}{\sum\limits_{m,m^{\prime}}{\int\limits_{\Omega}{d^{2}r_{0||}{G_{\mu m}\left( {r,{r_{0||};\omega}} \right)} \times {\int_{\Omega^{\prime}}{d^{2}r_{0||}^{\prime}{G_{m^{\prime}v}^{*}\left( {r,{r_{0||}^{\prime};\omega^{\prime}}} \right)}}}}}}} < {\delta{{\overset{\sim}{p}}_{m}\left( {r_{0},\omega} \right)}\delta{{\overset{\sim}{p}}_{m^{\prime}}^{*}\left( {r_{0^{\prime}},\omega} \right)}}>={\frac{\omega^{2}}{c^{4}ɛ_{0}^{2}}{\sum_{m}{\int_{\Omega}{d^{2}r_{0||}{G_{\mu m}\left( {r,{r_{0||};\omega}} \right)}{G_{m^{\prime}\nu}^{*}\left( {r^{\prime},{r_{0||};\omega^{\prime}}} \right)} \times {\Theta(\omega)}{\sigma_{{mm}^{\prime}}^{\prime 2D}\left( {r_{0||\prime}\omega} \right)}{\delta\left( {\omega - \omega^{\prime}} \right)}}}}}} = {\frac{\omega^{2}}{c^{4}ɛ_{0}^{2}}{\sum\limits_{m,j}{{G_{\mu m}\left( {r,{R_{j};\omega}} \right)}{G_{m\;\nu}^{*}\left( {r^{\prime},{R_{j};\omega^{\prime}}} \right)} \times {\Theta(\omega)}{\sigma_{mm}^{\prime 2D}\left( {R_{j},\omega} \right)}{\delta\left( {\omega - \omega^{\prime}} \right)}}}}} & (40)\end{matrix}$

noting that the dielectric tensor ε″(r,ω) is diagonal.

FIG. 7 shows spectral radiance diagram 525 of NPG including cavity, asshown in in FIGS. 1 and 2, as a function of wavelength λ, with Fermienergy E_(F)=−1.0 eV, mobility μ=3000 V/cm² s, hole diameter of a=90 nm,and period of

=150 nm at 1300 K, 1700 K, and 2000 K.

Since the energy density of the emitted electric field at the point r is

u(r,ω)δ(ω−ω′)=ε₀Σ_(i=x,y,z) <δÊ _(i)*(r,ω)δÊ _(i)(r,ω′)>,  (41)

we can write the spectral radiance as

$\begin{matrix}{{{I\left( {r,\omega} \right)} = {{\frac{\omega^{2}}{4\pi c^{3}ɛ_{0}}\frac{1}{N}{\sum\limits_{{\mu;{m = x}},{y;j}}{{{G_{\mu m}\left( {r,{R_{j};\omega}} \right)}}^{2} \times {\Theta(\omega)}{\sigma_{mm}^{\prime 2D}\left( {R_{j},\omega} \right)}}}} = {\frac{\omega^{2}}{4\pi c^{3}ɛ_{0}}{\Theta(\omega)}{\sigma_{||}^{\prime 2D}(\omega)}{\sum_{\mu,m}{{G_{\mu,m}\left( {r,{R_{0};\omega}} \right)}}^{2}}}}},} & (42)\end{matrix}$

assuming that the dipole current of the LSP is in the plane of thegraphene sheet, i.e. the xy-plane, and the polarizability is isotropic,i.e. σ′_(∥) ^(2D)=σ′_(xx) ^(2D)=σ′_(yy) ^(2D), and the same for allholes. N is the number of holes. In order to obtain the spectralradiance in the far field, we need to integrate over the sphericalangle. Using the results from the Supplementary Information, we obtain

$\begin{matrix}{{{I_{\infty}(\omega)} = {{\frac{\omega^{2}{\Theta(\omega)}}{3\pi^{2}ɛ_{0}c^{3}}{\sigma_{||}^{\prime 2D}(\omega)}} = {\frac{\omega^{2}{\Theta(\omega)}}{3c^{2}\pi^{2}}{A_{||}^{2D}(\omega)}}}},} & (43)\end{matrix}$

where we used the definition of the absorbance of a 2D material, i.e.

A _(2D)(ω)=(1/ε₀ c)Reσ _(2D)(ω)=(1/ε₀ c)σ_(2D)′(ω),  (44)

with 2D complex conductivity σ_(2D)(ω). According to Kirchhoff's law,emittance ϵ(ω), absorbance A(ω), reflectance R(ω), and transmittanceT(ω) are related by [16]

ϵ(ω)=A(ω)=1−R(ω)−T(ω),  (45)

from which we obtain the grey-body thermal emission formula

$\begin{matrix}{{{I_{\infty}(\omega)} = {\frac{\omega^{2}{\Theta(\omega)}}{3\pi^{2}c^{2}}{\epsilon_{||}^{2D}(\omega)}}},} & (46)\end{matrix}$

whose prefactor bears strong similarity to Planck's black body formulain Eq. (1).

FIGS. 8A-8B shows temperature distribution diagram 530-531 inside theNPG sheet for various values of the bias voltage V_(SD), calculated bymeans of COMSOL. As the bias voltage is increased, the maximum oftemperature shifts away from the center of the NPG sheet due to thePeltier effect.

Using FDTD to calculate the emittance ϵ_(∥) ^(2D)(ω), we evaluated thegrey-body thermal emission according to Eq. (46) for the thermal emitterstructure based on NPG shown in FIGS. 1 and 2. Using COMSOL, wecalculated the temperature distribution inside the NPG sheet, as shownin FIG. 7, when a bias voltage V_(SD) is applied, which gives rise toJoule heating. Our results are shown in FIGS. 5, 6, and 8 for thetemperatures 1300 K, 1700 K, and 2000 K of NPG. After integrating overthe wavelength under the curves, we obtain the following thermalemission power per area:

Resonance wavelength Power per area  4 μm 11,221 W/m²  7 μm   9820 W/m²10 μm   6356 W/m²

Let us consider the dependence of the thermal emission of NPG on theangle θ. Integrating over r²φ we obtain

$\begin{matrix}{{{I\left( {\theta,\omega} \right)} = {\frac{\omega^{2}}{4\pi c^{2}}{\Theta(\omega)}\frac{{11} + {\cos\left( {2\theta} \right)}}{16\pi}{\epsilon_{||}^{2D}(\omega)}}},} & (47)\end{matrix}$

which is a clear deviation from a Lambert radiator. The pattern of thethermal radiation can be determined by

$\begin{matrix}{{{\hat{I}(\theta)} = {\frac{\int_{0}^{2\pi}{{I\left( {r,\omega} \right)}r^{2}d\;\varphi}}{\int_{0}^{2\pi}{\int_{0}^{\pi}{{I\left( {r,\omega} \right)}r^{2}\sin\;\theta\; d\;\theta\; d\;\varphi}}} = {\frac{3}{64}\left\lbrack {{11} + {\cos\left( {2\theta} \right)}} \right\rbrack}}},} & (48)\end{matrix}$

which is shown in FIG. 9. Interestingly, since we assumed that thermalemission is completely incoherent [see Eq. (42)] the thermal emissionfrom NPG is only weakly dependent on the emission angle θ, which can beclearly seen in FIG. 9.

FIG. 9 shows spectral radiance diagram 535 of NPG including cavity, asshown in in FIGS. 1 and 2, as a function of wavelength λ with Fermienergy E_(F)=−1.0 eV, mobility μ=3000 V/cm² s, hole diameter of a=300nm, and period of

=450 nm at 1300 K, 1700 K, and 2000 K.

However, the assumption that thermal emission of radiation is incoherentis not always true. Since Kirchhoff's law is valid, thermal sources canbe coherent. [11] After theoretical calculations predicted thatlong-range coherence may exist for thermal emission in the case ofresonant surface waves, either plasmonic or phononic in nature, [5, 12,34] experiments showed that a periodic microstructure in the polarmaterial SiC exhibits coherence over many wavelengths and radiates inwell-defined and controlled directions. [10] Here we show that thecoherence length of a graphene sheet patterned with circular holes canbe as large as 150 μm due to the plasmonic wave in the graphene sheet,thereby paving the way for the creation of phased arrays made ofnanoantennas represented by the holes in NPG.

The coherence of thermal emission can be best understood by means of anonlocal response function. [13] First, we choose the nonlocalhydrodynamic response function in Eq. (13). Using the 2D version of thefluctuation-dissipation theorem in Eq. (33), we obtain the nonlocalfluctuation-dissipation theorem in the hydrodynamic approximation,

$\begin{matrix}{{{< {\delta{{\hat{J}}_{\mu}\left( {r_{||},\omega} \right)}\delta{{\hat{J}}_{\nu}\left( {r_{||}^{\prime},\omega^{\prime}} \right)}}>={{\sigma_{\mu\nu}^{HD}\left( {{\Delta r_{||}},\omega} \right)}{\Theta(\omega)}{\delta\left( {\omega - \omega^{\prime}} \right)}}} = {{\frac{1}{D}{\int\limits_{0}^{\infty}{dk_{||}\frac{{\sigma_{intra}(\omega)}e^{{- i}k_{||}\Delta r_{||}}}{1 - {\eta^{2}\frac{k_{||}^{2}}{\omega^{2}}}}{\Theta(\omega)}{\delta\left( {\omega - \omega^{\prime}} \right)}}}} = {{\sigma_{intra}(\omega)}\frac{\omega\sqrt{\pi/2}}{D\eta}{\sin\left( \frac{\omega\Delta r_{||}}{\eta} \right)}{\Theta(\omega)}{\delta\left( {\omega - \omega^{\prime}} \right)}}}},} & (49)\end{matrix}$

where Δr_(∥)=r_(∥)−r_(∥)′ and η²β²+D²ω(γ+iω). This result suggests thatthe coherence length is given approximately by D, which according to [2]would be D≈0.4 μm. However, the resulting broadening of the LSPresonance peaks would be very large and therefore in completecontradiction to the experimental measurements of the LSP resonancepeaks in Refs. [24], [35], [25]. Thus, we conclude that the hydrodynamicdiffusion length must be frequency-dependent with D(v=0)=0.4 μm. Usingthe Fermi velocity of v_(F)=10⁶ m/s and a frequency of v=30 THz, theaverage oscillation distance is about L=v_(F)v⁻¹=0.033 μm, which is muchsmaller than D(v=0) in graphene. Thus, we can approximate D(v=30 THz)=0.We conjecture that there is a crossover for D into the hydrodynamicregime when the frequency is reduced below around v₀=1 to 3 THz, belowwhich the hydrodynamic effect leads to a strong broadening of the LSPpeaks for NPG. Consequently, the viscosity of graphene should also befrequency-dependent and a crossover for the viscosity should happen atabout the same frequency v₀. We plan to elaborate this conjecture infuture work. Future experiments could corroborate our conjecture bymeasuring the absorbance or emittance as a function of wavelength forvarying scale of patterning of the graphene sheet.

FIG. 10 shows a spherical density plot diagram 540 of the normalizedangular intensity distribution Î(θ) of the thermal emission from NPG inthe case of incoherent photons.

Next, let us consider the coherence of thermal emission by means of thenonlocal optical conductivity in the RPA approximation. Using thegeneral formula

$\begin{matrix}{{{\sigma\left( {q,\omega} \right)} = {\frac{{ie}^{2}\omega}{q^{2}}{\chi^{0}\left( {q,\omega} \right)}}},} & (50) \\{with} & \; \\{{\chi^{0}\left( {q,\omega} \right)} \approx \frac{\epsilon_{F}q^{2}}{\pi\hslash^{2}{\omega\left( {\omega + {i\;\tau^{- 1}}} \right)}}} & (51)\end{matrix}$

in the low-temperature and low-frequency approximation, one obtains Eq.(12). Now, let us use the full polarization in RPA approximationincluding only the Coulomb interaction,

$\begin{matrix}{{{\chi^{RPA}\left( {q,\omega} \right)} = \frac{\chi^{0}\left( {q,\omega} \right)}{1 - {{v_{c}(q)}{\chi^{0}\left( {q,\omega} \right)}}}},} & (52)\end{matrix}$

from which we obtain

$\begin{matrix}{{{\sigma^{RPA}\left( {q,\omega} \right)} = {{\frac{ie^{2}\omega}{q^{2}}{\chi\left( {q,\omega} \right)}} = \frac{{ie}^{2}{\omega\epsilon}_{F}}{{\pi\hslash^{2}{\omega\left( {\omega + {i\;\tau^{- 1}}} \right)}} - {\frac{e^{2}\epsilon_{F}}{2\epsilon_{0}}q}}}},} & (53)\end{matrix}$

which introduces the nonlocal response via the Coulomb interaction inthe denominator. The effect of the RPA correction on the LSP resonancesat around λ=4 μm, 7 μm, and 10 μm is shown in FIGS. 2, 3, and 4,respectively. After taking the Fourier transform, we obtain the nonlocalfluctuation-dissipation theorem in RPA approximation,

$\begin{matrix}{{{< {\delta{{\overset{\hat{}}{J}}_{\mu}\left( {r_{||},\omega} \right)}\delta{{\overset{\hat{}}{J}}_{v}\left( {r_{||}^{\prime},\omega^{\prime}} \right)}}>={{\sigma_{\mu\; v}^{RPA}\left( {{\Delta r_{||}},\omega} \right)}{\Theta(\omega)}{\delta\left( {\omega - \omega^{\prime}} \right)}}} = {\frac{\sqrt{2\pi}ɛ_{0}\omega}{c_{RPA}}e^{{iK}_{RPA}\Delta\; r_{||}\frac{\Delta\; r_{||}}{c_{RPA}}}{\Theta(\omega)}{\delta\left( {\omega - \omega^{\prime}} \right)}}},} & (54)\end{matrix}$

where the coherence length in RPA approximation is

$\begin{matrix}{{C_{RPA} = \frac{e^{2}{\epsilon_{F}}}{2{\pi\hslash}^{2}\epsilon_{0}\gamma\;\omega}},} & (55)\end{matrix}$

and the coherence wavenumber is given by

$\begin{matrix}{{K_{RPA} = \frac{2\pi\hslash^{2}\epsilon_{0}\omega^{2}}{e^{2}{\epsilon_{F}}}}.} & (55)\end{matrix}$

For simplicity, we switch now to a square lattice of holes. In the caseof the LSP resonance for a square lattice of holes at λ=10 μm,corresponding to v=30 THz, ϵ_(F)=−1.0 eV, ω=2 πv, and γ=ev_(F)²/(μE_(F))=0.3 THz for μ=3000 cm²V⁻¹ s⁻¹, which results in a coherencelength of C_(RPA)=3 μm. This result is in reasonable agreement with thefull width at half maximum (FWHM) values of the widths of the LSPresonance peaks in Refs. [24], [35], [25]. This coherence length wouldallow to preserve coherence for a linear array of period

=300 nm and C_(RPA)/

=10 holes. In order to show the coherence length that can be achievedwith graphene, we can consider a suspended graphene sheet with amobility of μ=15000 cm²V⁻¹ s⁻¹. Then the coherence length increases to avalue of C_(RPA)=13 μm, which would allow for coherence over a lineararray with C_(RPA)/

=43 holes.

FIG. 11 shows a diagram 545 with a coherence length C_(FDTD) andcoherence time τ_(FDTD) of emitted photons, extracted from thefull-width half-maximum (FWHM) of the spectral radiances shown in FIGS.5, 6, and 8. In the case of the LSP resonance for a square lattice ofholes at λ=5 μm, corresponding to v=60 THz, ϵ_(F)=−1.0 eV, ω=2 πv, andγ=ev_(F) ²/(μE_(F))=0.3 THz for μ=3000 cm² V⁻¹ s⁻¹, which results in acoherence length of C_(RPA)=1.5 μm. Considering again a suspendedgraphene sheet, the coherence length can be increased to C_(RPA)=6.7 μm.Since the period in this case is

=45 nm, the coherence for μ=3000 cm²V⁻¹ s⁻¹ and μ=15000 cm²V⁻¹ s⁻¹ canbe preserved for a linear array of C_(RPA)/

=33 and 148 holes, respectively.

The coherence length and time of thermally emitted photons is largerbecause the photons travel mostly in vacuum. Taking advantage of theWiener-Kinchine theorem, [11] we can extract the coherence lengthC_(FDTD) and coherence time τ_(FDTD) of thermally emitted photons bymeans of the full-width half-maximum (FWHM) of the spectral radiancesshown in FIGS. 5, 6, and 8. Our results are shown in FIG. 10. Thecoherence length of the thermally emitted photons can reach up toC_(FDTD)=150 μm at a resonance wavelength of λ=4 μm. This means that thecoherence length of the thermally emitted photons is about 37 timeslarger than the wavelength.

Thus, the latter large coherence length allows for the coherent controlof a 150×150 square array of holes with period

=45 nm, individually acting as nanoantennas, that can be used to createa phased array of nanoantennas. One of the intriguing properties of aphased array is that it allows to control the directivity of theemission of photons, which is currently being implemented for large 5 Gantennas in the 3 to 30 GHz range. The beamsteering capability of ourNPG sheet is shown in FIG. 11. In contrast, our proposed phased arraybased on NPG can operate in the 10 to 100 THz range.

The temporal control of the individual phases of the holes requires anextraordinary fast switching time of around 1 ps, which is not feasiblewith current electronics. However, the nonlocal response functionreveals a spatial phase shift determined by the coherence wavenumberK_(RPA), which is independent of the mobility of graphene. In the caseof the LSP resonance at λ=4 μm, we obtain λ_(RPA)=2π/K_(RPA)=6 μm,resulting in a minimum phase shift of 2 π

/λ_(RPA)=0.042=2.4° between neighboring holes, which can be increased toa phase shift of 9.7° by decreasing the Fermi energy to E_(F)=−0.25 eV.Thus, the phase shift between neighboring holes can be tuned arbitrarilybetween 2.4° and 9.7° by varying the Fermi energy between ϵ_(F)=−1.0 eVand ϵ_(F)=−0.25 eV. FIG. 11 shows the capability of beamsteering for ourproposed structure by means of directional thermal emission, which istunable by means of the gate voltage applied to the NPG sheet.

Due to the full control of directivity with angle of emission betweenθ=12° and θ=80° by tuning the Fermi energy in the range betweenϵ_(F)=−1.0 eV and ϵ_(F)=−0.25 eV, thereby achieving beamsteering bymeans of the gate voltage, our proposed mid-IR light source based on NPGcan be used not only in a vertical setup for surface emission, but alsoin a horizontal setup for edge emission, which is essential fornanophotonics applications.

FIGS. 12A-12D show diagram 550-553 for directivity of the thermalemission from NPG where the holes act as nanoantennas in a phased array.This emission pattern for ϵ_(F)=−1.0 eV can be used for surface-emittingmid-IR sources. In the case of a 150×150 (diagram 550), 75×75 (diagram551), 56×56 (diagram 552), 37×37 (diagram 553) square lattice of holes(size of lattice matches coherence length) with period

=45 nm and hole diameter of 30 nm, introducing a relative phase of2.43°, 4.8 6°, 7.28°, 9.71° between the nanoantennas allows forbeamsteering in the range between θ=12° (diagram 550) and θ=80° (diagram553) by tuning the Fermi energy in the range between ϵ_(F)=−1.0 eV andϵ_(F)=−0.25 eV.

In conclusion, we have demonstrated in our theoretical study that NPGcan be used to develop a plasmonically enhanced mid-IR light source withspectrally tunable selective thermal emission. Most importantly, theLSPs along with an optical cavity increase substantially the emittanceof graphene from about 2% for pristine graphene to 80% for NPG, therebyoutperforming state-of-the-art graphene light sources working in thevisible and NIR by at least a factor of 100. Combining our proposedmid-IR light source based on patterned graphene with our demonstratedmid-IR detector based on NPG [25], we are going to develop a mid-IRspectroscopy and detection platform based on patterned graphene thatwill be able to detect a variety of molecules that have mid-IRvibrational resonances, such as CO, CO₂, NO, NO₂, CH₄, TNT, H₂O₂,acetone, TATP, Sarin, VX, etc. In particular, a recent study showed thatit is possible to detect the hepatitis B and C viruses label-free at awavelength of around 6 μm. [23] Therefore, we will make great effort todemonstrate that our platform will be able to detect with highsensitivity and selectivity the COVID-19 virus and other viruses thatpose a threat to humanity.

FIG. 13 shows a diagram 555 of theoretical fit to spectral radiancepresented in [26]. Shiue et al. [26] used a photonic crystal structureto filter the thermal emission from pristine graphene with an emittanceof around A=0.07. Integrating the spectral radiance under the curvegives a value of about P/A=100 W/m², which is about 100 times weakerthan our proposed thermal radiation source based on NPG.

Kirchhoff's law of thermal radiation states that emittance ϵ is equal toabsorbance A, i.e.

ϵ(ω,θ,ϕ,T)=A(ω,θ,ϕ,T).  (57)

In the case of a black body ϵ(ω,θ,ϕ,T)=A(ω,θ,ϕ,T)=1. Pristine graphenehas a very small absorbance of only A=0.023 and is a nearly transparentbody. Shiue et al. [26] used a photonic crystal structure to filter thethermal emission from pristine graphene with an emittance of aroundA=0.07. [26] Their spectral radiance is shown in FIGS. 12A-12D andexhibits peaks at around λ=1.55 μm at a temperature of T=2000 K. Afterintegrating the spectral radiance under the curve, one obtains anemission power per area of about P/A=100 W/m², which is about 100 timesweaker than our proposed thermal radiation source based on NPG at T=2000K. Our proposed thermal mid-IR source features an emission power perarea of about P/A=10⁴ W/m² at T=2000 K. In addition, our proposedthermal mid-IR source features frequency-tunability and beamsteering bymeans of a gate voltage applied to the NPG sheet.

FIG. 14 includes a diagram 560 showing that the NPG sheet allows forspectrally selective thermal emission at around λ=4.5 μm for a period of

=45 nm and a hole diameter of a=30 nm. Using FDTD to calculate theemittance ϵ_(∥) ^(2D)(ω), we evaluated the grey-body thermal emissionaccording to Eq. (46) for the thermal emitter structure based on NPGshown in FIGS. 1 and 2. Our results for the temperature T=300 K of NPGare shown in FIGS. 13, 14, and 15. In these figures we compare ourresults for NPG with the results for pristine graphene and black bodyradiation. FIG. 15 includes a diagram 565 showing that the NPG sheetallows for spectrally selective thermal emission at around λ=7 μm for aperiod of

=150 nm and a hole diameter of a=90 nm.

For determining the EM properties of an infinitesimally thin conductingelliptical disk of radius R or an infinitesimally thin conducting planewith a elliptical hole, including coated structures, it is mostconvenient to perform the analytical calculations in the ellipsoidalcoordinate system (ξ, η, ζ), [1, 18, 15, 4] which is related to theCartesian coordinate system through the implicit equation

$\begin{matrix}{{\frac{x^{2}}{a^{2} + u} + \frac{y^{2}}{b^{2} + u} + \frac{z^{2}}{c^{2} + u}} = 1} & (58)\end{matrix}$

for a>b>c. The cubic roots ξ, η, and ζ are all real in the ranges

−a ² ≤ζ≤−b ² ,−b ² ≤η≤−c ² ,−c ²≤ξ<∞,  (59)

which are the ellipsoidal coordinates of a point (x, y, z). The surfacesof contant ξ, η, and ζ are ellipsoids, hyperboloids of one sheet, andhyperboloids of two sheets, respectively, all confocal with theellipsoid defined by

$\begin{matrix}{{{\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}}} = 1}.} & (60)\end{matrix}$

Each point (x, y, z) in space is determined by the intersection of threesurfaces, one from each of the three families, and the three surfacesare orthogonal to each other. The transformation between the twocoordinate systems is given by the solutions of Eq. (58), i.e.

$\begin{matrix}{{x = {\pm \sqrt{\frac{\left( {\xi + a^{2}} \right)\left( {\eta + a^{2}} \right)\left( {\zeta + a^{2}} \right)}{\left( {b^{2} - a^{2}} \right)\left( {c^{2} - a^{2}} \right)}}}},} & (61) \\{{y = {\pm \sqrt{\frac{\left( {\xi + b^{2}} \right)\left( {\eta + b^{2}} \right)\left( {\zeta + b^{2}} \right)}{\left( {c^{2} - b^{2}} \right)\left( {a^{2} - b^{2}} \right)}}}},} & (62) \\{{z = {\pm \sqrt{\frac{\left( {\xi + c^{2}} \right)\left( {\eta + c^{2}} \right)\left( {\zeta + c^{2}} \right)}{\left( {a^{2} - c^{2}} \right)\left( {b^{2} - c^{2}} \right)}}}},} & (63)\end{matrix}$

defining 8 equivalent octants. The length elements in ellipsoidalcoordinates read

$\begin{matrix}{{{dl}^{2} = {{h_{1}^{2}d\;\xi^{2}} + {h_{2}^{2}d\eta^{2}} + {h_{3}^{2}d\zeta^{2}}}},} & (64) \\{{h_{1} = \sqrt{\frac{\left( {\xi - \eta} \right)\left( {\xi - \zeta} \right)}{2R_{\xi}}}},} & (65) \\{{h_{2} = \sqrt{\frac{\left( {\eta - \zeta} \right)\left( {\xi - \zeta} \right)}{2R_{\eta}}}},} & (66) \\{{h_{3} = \sqrt{\frac{\left( {\zeta - \xi} \right)\left( {\zeta - \eta} \right)}{2R_{\zeta}}}},} & (67) \\{{R_{u}^{2} = {\left( {u + a^{2}} \right)\left( {u + b^{2}} \right)\left( {u + c^{2}} \right)}},{u = \xi},\eta,{\zeta.}} & (68)\end{matrix}$

For the transformation from Cartesian to ellipsoidal coordinates, onecan use the following system of equations:

$\begin{matrix}{{\xi = \frac{{\frac{\partial x}{\partial\xi}x} + {\frac{\partial y}{\partial\xi}y} + {\frac{\partial z}{\partial\xi}z}}{\sqrt{\left( \frac{\partial x}{\partial\xi} \right)^{2} + \left( \frac{\partial y}{\partial\xi} \right)^{2} + \left( \frac{\partial z}{\partial\xi} \right)^{2}}}},} & (69) \\{{\eta = \frac{{\frac{\partial x}{\partial\eta}x} + {\frac{\partial y}{\partial\eta}y} + {\frac{\partial z}{\partial\eta}z}}{\sqrt{\left( \frac{\partial x}{\partial\xi} \right)^{2} + \left( \frac{\partial y}{\partial\xi} \right)^{2} + \left( \frac{\partial z}{\partial\xi} \right)^{2}}}},} & (70) \\{{\zeta = \frac{{\frac{\partial x}{\partial\eta}x} + {\frac{\partial y}{\partial\eta}y} + {\frac{\partial z}{\partial\eta}z}}{\sqrt{\left( \frac{\partial x}{\partial\zeta} \right)^{2} + \left( \frac{\partial y}{\partial\zeta} \right)^{2} + \left( \frac{\partial z}{\partial\zeta} \right)^{2}}}},} & (71)\end{matrix}$

whose elements J_(ij) define the Jacobian matrix. The derivatives areexplicitly:

$\begin{matrix}{{\frac{\partial x}{\partial\xi} = {\frac{1}{2}\sqrt{\frac{\left( {a^{2} + \eta} \right)\left( {a^{2} + \zeta} \right)}{\left( {a^{2} + \xi} \right)\left( {a^{2} - b^{2}} \right)\left( {a^{2} - c^{2}} \right)}}}},} & (72) \\{{\frac{\partial x}{\partial\eta} = {\frac{1}{2}\sqrt{\frac{\left( {a^{2} + \xi} \right)\left( {a^{2} + \zeta} \right)}{\left( {a^{2} + \eta} \right)\left( {a^{2} - b^{2}} \right)\left( {a^{2} - c^{2}} \right)}}}},} & (73) \\{{\frac{\partial x}{\partial\zeta} = {\frac{1}{2}\sqrt{\frac{\left( {a^{2} + \xi} \right)\left( {a^{2} + \eta} \right)}{\left( {a^{2} + \zeta} \right)\left( {a^{2} - b^{2}} \right)\left( {a^{2} - c^{2}} \right)}}}},} & (74) \\{{\frac{\partial y}{\partial\xi} = {\frac{1}{2}\sqrt{\frac{\left( {b^{2} + \eta} \right)\left( {b^{2} + \zeta} \right)}{\left( {b^{2} + \xi} \right)\left( {b^{2} - a^{2}} \right)\left( {b^{2} - c^{2}} \right)}}}},} & (75) \\{{\frac{\partial y}{\partial\eta} = {\frac{1}{2}\sqrt{\frac{\left( {b^{2} + \xi} \right)\left( {b^{2} + \zeta} \right)}{\left( {b^{2} + \eta} \right)\left( {b^{2} - a^{2}} \right)\left( {b^{2} - c^{2}} \right)}}}},} & (76) \\{{\frac{\partial y}{\partial\zeta} = {\frac{1}{2}\sqrt{\frac{\left( {b^{2} + \xi} \right)\left( {b^{2} + \eta} \right)}{\left( {b^{2} + \zeta} \right)\left( {b^{2} - a^{2}} \right)\left( {b^{2} - c^{2}} \right)}}}},} & (77) \\{{\frac{\partial z}{\partial\xi} = {\frac{1}{2}\sqrt{\frac{\left( {c^{2} + \eta} \right)\left( {c^{2} + \zeta} \right)}{\left( {c^{2} + \xi} \right)\left( {c^{2} - a^{2}} \right)\left( {c^{2} - b^{2}} \right)}}}},} & (78) \\{{\frac{\partial z}{\partial\eta} = {\frac{1}{2}\sqrt{\frac{\left( {c^{2} + \xi} \right)\left( {c^{2} + \zeta} \right)}{\left( {c^{2} + \eta} \right)\left( {c^{2} - a^{2}} \right)\left( {c^{2} - b^{2}} \right)}}}},{\frac{\partial z}{\partial\zeta} = {\frac{1}{2}{\sqrt{\frac{\left( {c^{2} + \xi} \right)\left( {c^{2} + \eta} \right)}{\left( {c^{2} + \zeta} \right)\left( {c^{2} - a^{2}} \right)\left( {c^{2} - b^{2}} \right)}}.}}}} & (79)\end{matrix}$

FIG. 16 includes a diagram 570 showing that the NPG sheet allows forspectrally selective thermal emission at around λ=10 μm for a period of

=450 nm and a hole diameter of a=300 nm. The coordinate η is constant onthe surfaces of oblate spheroids defined by

$\begin{matrix}{{\frac{x^{2} + y^{2}}{\left( {R\;\cosh\;\eta} \right)^{2}} + \frac{z^{2}}{\left( {R\sinh\eta} \right)^{2}}} = 1} & (81)\end{matrix}$

The surface associated with the limit η→0 is an infinitesimally thincircular disk of radius R. In contrast, the surface in the limit η>>1 isa sphere of radius r=R cos hη≈R sin hη. Thus, the Laplace equation inellipsoidal coordinates reads

$\begin{matrix}{{\Delta\Phi} = {{\frac{4}{\left( {\xi - \eta} \right)\left( {\zeta - \xi} \right)\left( {\eta - \zeta} \right)}\left\lbrack {{\left( {\eta - \zeta} \right)R_{\xi}\frac{\partial}{\partial\xi}\left( {R_{\xi}\frac{\partial\Phi}{\partial\xi}} \right)} + {\left( {\zeta - \xi} \right)R_{\eta}\frac{\partial}{\partial\eta}\left( {R_{\eta}\frac{\partial\Phi}{\partial\eta}} \right)} + {\left( {\xi - \eta} \right)R_{\zeta}\frac{\partial}{\partial\zeta}\left( {R_{\zeta}\frac{\partial\Phi}{\partial\zeta}} \right)}} \right\rbrack} = 0.}} & (82)\end{matrix}$

The surface of the conducting ellipsoid is defined by ξ=0. Thus, theelectric field potential Φ(ξ) is a function of ξ only, thereby definingthe equipotential surfaces by confocal ellipsoids. Laplace's equation isthen simplified to

$\begin{matrix}{{{\frac{d}{d\xi}\left( {R_{\xi}\frac{d\Phi}{d\xi}} \right)} = 0}.} & (83)\end{matrix}$

The solution outside the ellipsoid is

$\begin{matrix}{{\Phi_{out}(\xi)} = {A{\int_{\xi}^{\infty}{\frac{d\;\xi^{\prime}}{R_{\xi^{\prime}}}.}}}} & (84)\end{matrix}$

From the asymptotic approximation ξ≈r² for large distances r→∞, i.e.ξ→∞, we identify R_(ξ)≈ξ^(3/2) and thus

$\begin{matrix}{{{\Phi_{out}\left( \xi\rightarrow\infty \right)} \approx \frac{2A}{\sqrt{\xi}}} = {\frac{2A}{r}.}} & (85)\end{matrix}$

using the boundary condition lim_(ξ→∞)Φ(ξ)=0. since the Coulomb fieldshould be Φ(ξ→∞)≈e/r at large distances from the ellipsoid, 2A=e and

$\begin{matrix}{{\Phi_{out}(\xi)} = {\frac{e}{2}{\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{R_{\xi^{\prime}}}}}} & (86)\end{matrix}$

is obtained, corresponding to the far-field of a monopole charge.

The solution inside the ellipsoid is

$\begin{matrix}{{\Phi_{in}(\xi)} = {B{\int_{- c^{2}}^{\xi}{\frac{d\;\xi^{\prime}}{R_{\xi^{\prime}}}.}}}} & (87)\end{matrix}$

Using the asymptotic approximation R_(ξ→-c) ₂ ∝√{square root over(ξ+c²)} we obtain

Φ_(in)(ξ→−c ²)≈B√{square root over (ξ+c ²)}.  (88)

This solution satisfies the boundary condition lim_(ξ→-c) ₂ Φ(ξ)=0. Theconstant B can be found from the boundary condition Φ(ξ=0)=V, where V isthe potential on the surface of the charged ellipsoid. Thus, B=V/c and

$\begin{matrix}{{{\Phi_{in}(\xi)} = {\frac{V}{c}\sqrt{\xi + c^{2}}}}.} & (89)\end{matrix}$

Following Ref. Bohren 1998, let us consider the case when the externalelectric field is parallel to one of the major axes of the ellipsoid.For the external potential let us choose

$\begin{matrix}{\Phi_{0} = {{{- E_{0}}z} = {{- E_{0}}\sqrt{\frac{\left( {\xi + c^{2}} \right)\left( {\eta + c^{2}} \right)\left( {\zeta + c^{2}} \right)}{\left( {a^{2} - c^{2}} \right)\left( {b^{2} - c^{2}} \right)}}}}} & (90)\end{matrix}$

Let Φ_(p) be the potential caused by the ellipsoid, with the boundarycondition Φ_(p)(ξ→∞)=0. Requiring continuous boundary condition on thesurface of the ellipsoid, we have

Φ_(in)(0,η,ζ)=Φ₀(0,η,ζ)+Φ_(p)(0,η,ζ).  (91)

We make the ansatz

Φ_(p)(ξ,η,ζ)=F _(p)(ξ)√{square root over ((η+c ²)(ζ+c ²))},  (92)

which after insertion into the Laplace equation yields

$\begin{matrix}{{{{R_{\xi}{\frac{d}{d\xi}\left\lbrack {R_{\xi}\frac{dF}{d\xi}} \right\rbrack}} - {\left( {\frac{a^{2} + b^{2}}{4} + \frac{\xi}{2}} \right){F(\xi)}}} = 0}.} & (93)\end{matrix}$

Thus, one obtains for the field caused by the ellipsoid

Φ_(p)(ξ,η,ζ)=C _(p) F _(p)(ξ)√{square root over ((η+c ²)(ζ+c ²))}  (94)

with

$\begin{matrix}{{{F_{p}(\xi)} = {{F_{in}(\xi)}{\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{{F_{in}^{2}\left( \xi^{\prime} \right)}R_{\xi^{\prime}}}}}},} & (95)\end{matrix}$

where

F _(in)(ξ)=√{square root over (ξ+c ²)},  (96)

the function we used in the case of the charged ellipsoid (see above).Thus, the field inside the ellipsoid is given by

Φ_(in) =C _(in) F _(in)(ξ)√{square root over ((η+c ²)(ζ+c ²))}.  (97)

Using the boundary condition shown in Eq. (91), one obtains the firstequation

$\begin{matrix}{{{{C_{p}{\int_{0}^{\infty}\frac{d\;\xi^{\prime}}{\left( {c^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}}} - C_{in}} = \frac{E_{0}}{\sqrt{\left( {a^{2} - c^{2}} \right)\left( {b^{2} - c^{2}} \right)}}},} & (98)\end{matrix}$

The boundary condition of the normal component of D at ξ=0, equivalentto

$\begin{matrix}{{{ɛ_{in}\frac{\partial\Phi_{in}}{\partial\xi}} = {{ɛ_{m}\frac{\partial\Phi_{0}}{\partial\xi}} + {ɛ_{m}\frac{\partial\Phi_{p}}{\partial\xi}}}},} & (99)\end{matrix}$

yields the second equation

$\begin{matrix}{{{ɛ_{m}{C_{p}\left\lbrack {{\int\limits_{0}^{\infty}\frac{d\;\xi^{\prime}}{\left( {c^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}} - \frac{2}{abc}} \right\rbrack}} - {ɛ_{in}C_{in}}} = {\frac{ɛ_{m}E_{0}}{\sqrt{\left( {a^{2} - c^{2}} \right)\left( {b^{2} - c^{2}} \right)}}.}} & (100)\end{matrix}$

Consequently, the potentials are

$\begin{matrix}{{\Phi_{in} = \frac{\Phi_{0}}{1 + \frac{L_{3}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}}\ ,} & (101) \\{\Phi_{p} = {\Phi_{0}\frac{\frac{abc}{2}\frac{ɛ_{m} - ɛ_{in}}{ɛ_{m}}{\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{\left( {c^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}}}{1 + \frac{L_{3}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}}} & (102) \\{where} & \; \\{L_{3} = {\frac{abc}{2}{\int_{0}^{\infty}{\frac{d\;\xi^{\prime}}{\left( {c^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}.}}}} & (103)\end{matrix}$

Far away from the ellipsoid for ξ≈r²→∞, one can use the approximation

$\begin{matrix}{{{{\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{\left( {c^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}} \approx {\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{\xi^{{\prime 5}/2}}}} = {\frac{2}{3}\xi^{{- 3}/2}}},} & (104)\end{matrix}$

yielding the potential caused by the ellipsoid, i.e.

$\begin{matrix}{{\Phi_{p} \approx {\frac{E_{0}\cos\;\theta}{r^{2}}\frac{\frac{abc}{2}\frac{ɛ_{in} - ɛ_{m}}{ɛ_{m}}}{1 + \frac{L_{3}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}}},} & (105)\end{matrix}$

from which we identify the dipole moment

$\begin{matrix}{p = {{pz} = {4\pi ɛ_{m}{abc}\frac{ɛ_{in} - ɛ_{m}}{{3ɛ_{m}} + {3{L_{3}\left( {ɛ_{in} - ɛ_{m}} \right)}}}E_{0}{z.}}}} & (106)\end{matrix}$

This result determines the polarizability of the charged ellipsoid, i.e.

$\begin{matrix}{\alpha_{3} = {4{\pi ɛ}_{m}{abc}\frac{ɛ_{in} - ɛ_{m}}{{3ɛ_{m}} + {3{L_{3}\left( {ɛ_{in} - ɛ_{m}} \right)}}}}} & (107)\end{matrix}$

If the external electric field is applied along the other major axes ofthe ellipsoid, x or y, the polarizabilities are

$\begin{matrix}{{\alpha_{1} = {4{\pi ɛ}_{m}{abc}\frac{ɛ_{in} - ɛ_{m}}{{3ɛ_{m}} + {3{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}}}}},} & (108) \\{{\alpha_{2} = {4{\pi ɛ}_{m}{abc}\frac{ɛ_{in} - ɛ_{m}}{{3ɛ_{m}} + {3{L_{2}\left( {ɛ_{in} - ɛ_{m}} \right)}}}}},} & (109)\end{matrix}$

respectively, where

$\begin{matrix}{{L_{1} = {\frac{abc}{2}{\int_{0}^{\infty}\frac{d\;\xi^{\prime}}{\left( {a^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}}}},} & (110) \\{L_{2} = {\frac{abc}{2}{\int_{0}^{\infty}\frac{d\;\xi^{\prime}}{\left( {b^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}}}} & (111)\end{matrix}$

For oblate spheroids (a=b), L₁=L₂,

$\begin{matrix}{{L_{1} = {{\frac{g\left( e_{0} \right)}{2e_{o}^{2}}\left\lbrack {\frac{\pi}{2} - {\arctan\;{g\left( e_{o} \right)}}} \right\rbrack} - \frac{g^{2}\left( e_{o} \right)}{2}}},{{g\left( e_{0} \right)} = \sqrt{\frac{1 - e_{o}^{2}}{e_{o}^{2}}}},{e_{o}^{2} = {1 - \frac{c^{2}}{a^{2}}}},} & (112)\end{matrix}$

where e_(o) is the eccentricity of the oblate spheroid. The limitingcases of an infinitesimally thin disk and a sphere are obtained fore_(o)=1 and e_(o)=0, respectively.

The geometrical factors L_(i) are related to the depolarization factors{circumflex over (L)}_(i) by

$\begin{matrix}{{E_{inx} = {E_{0x} - {{\hat{L}}_{1}P_{inx}}}},} & (113) \\{{E_{iny} = {E_{0y} - {{\hat{L}}_{2}P_{iny}}}},} & (114) \\{{E_{inz} = {E_{0z} - {{\hat{L}}_{3}P_{inz}}}},{with}} & (115) \\{{\hat{L}}_{i} = {\frac{ɛ_{in} - ɛ_{m}}{ɛ_{in} - ɛ_{0}}{\frac{L_{i}}{ɛ_{m}}.}}} & (116)\end{matrix}$

In analogy to Ref. [4], let us consider the case when the externalelectric field is parallel to one of the major axes of the ellipsoid, inthis case along the x-axis. For the external potential let us choose

$\begin{matrix}{\Phi_{0} = {{{- E_{0}}x} = {{- E_{0}}{\sqrt{\frac{\left( {\xi + a^{2}} \right)\left( {\eta + a^{2}} \right)\left( {\zeta + a^{2}} \right)}{\left( {b^{2} - a^{2}} \right)\left( {c^{2} - a^{2}} \right)}}.}}}} & (117)\end{matrix}$

Let Φ_(p) be the potential caused by the ellipsoid, with the boundarycondition Φ_(p)(ξ→∞)=0. Requiring continuous boundary condition on thesurface of the ellipsoid, we have

Φ_(in)(0,η,ζ)=Φ₀(0,η,ζ)+Φ_(p)(0,η,ζ).  (118)

Thus, one obtains for the field caused by the ellipsoid

Φ_(p)(ξ,η,ζ)=C _(p) F _(p)(ξ)√{square root over ((η+a ²)(ζ+a ²))}  (119)

with

$\begin{matrix}{{{F_{p}(\xi)} = {{F_{in}(\xi)}{\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{{F_{in}^{2}\left( \xi^{\prime} \right)}R_{\xi^{\prime}}}}}},} & (120)\end{matrix}$

where

F _(in)(ξ)=√{square root over (ξ+a ²)},  (121)

the function we used in the case of the charged ellipsoid (see above).Thus, the field inside the ellipsoid is given by

Φ_(in) =C _(in) F _(in)(ξ)√{square root over ((η+a ²)(ζ+a ²))}.  (122)

Using the boundary condition shown in Eq. (118), one obtains the firstequation

$\begin{matrix}{{{{C_{p}{\int_{0}^{\infty}\frac{d\;\xi^{\prime}}{\left( {a^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}}} - C_{in}} = \frac{E_{0}}{\sqrt{\left( {b^{2} - a^{2}} \right)\left( {c^{2} - a^{2}} \right)}}},} & (123)\end{matrix}$

The boundary condition of the normal component of D at ξ=0, equivalentto

$\begin{matrix}{{{ɛ_{in}\frac{\partial\Phi_{in}}{\partial\xi}} = {{ɛ_{m}\frac{\partial\Phi_{0}}{\partial\xi}} + {ɛ_{m}\frac{\partial\Phi_{p}}{\partial\xi}}}},} & (124)\end{matrix}$

yields the second equation

$\begin{matrix}{{{ɛ_{m}{C_{p}\left\lbrack {{\int\limits_{0}^{\infty}\frac{d\;\xi^{\prime}}{\left( {a^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}} - \frac{2}{abc}} \right\rbrack}} - {ɛ_{in}C_{in}}} = {\frac{ɛ_{m}E_{0}}{\sqrt{\left( {b^{2} - a^{2}} \right)\left( {c^{2} - a^{2}} \right)}}.}} & (125)\end{matrix}$

Consequently, the potentials are

$\begin{matrix}{{\Phi_{in} = \frac{\Phi_{0}}{1 + \frac{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}},} & (126) \\{{\Phi_{p} = {\Phi_{0}\frac{\frac{abc}{2}\frac{ɛ_{m} - ɛ_{in}}{ɛ_{m}}{\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{\left( {a^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}}}{1 + \frac{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}}},} & (127)\end{matrix}$

where

$\begin{matrix}{L_{1} = {\frac{abc}{2}{\int_{0}^{\infty}{\frac{d\;\xi^{\prime}}{\left( {a^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}.}}}} & (128)\end{matrix}$

Far away from the ellipsoid for ξ≈r²→∞, one can use the approximation

$\begin{matrix}{{{{\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{\left( {a^{2} + \xi^{\prime}} \right)R_{\xi^{\prime}}}} \approx {\int_{\xi}^{\infty}\frac{d\;\xi^{\prime}}{\xi^{{\prime 5}\text{/}2}}}} = {\frac{2}{3}\xi^{{- 3}\text{/}2}}},} & (129)\end{matrix}$

yielding the potential caused by the ellipsoid, i.e.

$\begin{matrix}{{\Phi_{p} \approx {\frac{E_{0}\cos\;\theta}{r^{2}}\frac{\frac{abc}{3}\frac{ɛ_{in} - ɛ_{m}}{ɛ_{m}}}{1 + \frac{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}}},} & (130)\end{matrix}$

from which we identify the dipole moment

$\begin{matrix}{p = {{px} = {4{\pi ɛ}_{m}{abc}\frac{ɛ_{in} - ɛ_{m}}{{3ɛ_{m}} + {3{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}}}E_{0}{x.}}}} & (131)\end{matrix}$

This result determines the polarizability of the charged ellipsoid, i.e.

$\begin{matrix}{\alpha_{1} = {4{\pi ɛ}_{m}{abc}\frac{ɛ_{in} - ɛ_{m}}{{3ɛ_{m}} + {3{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}}}}} & (132)\end{matrix}$

Let us consider a conducting single-sheet hyperboloid with a smallelliptical wormhole, as shown in FIG. 16. Contrary to the case of anuncharged ellipsoid, where the solutions when applying the externalelectric field in x, y, or z direction are similar, the solutions in thecase of an uncharged hyperboloid depend strongly on the axis in whichthe external field E₀ points. While the solutions for E₀=E₀x and E₀=E₀yare similar, the solution for E₀=E₀z is completely different. The reasonfor this fundamental difference is that the ellipsoid resembles a spherefrom far away. However, a single-sheet hyperboloid has ellipticalcylindrical symmetry. FIG. 17 shows a schematic with a single-sheethyperboloid with an elliptical wormhole of length a, width b, and depthc=0. The electric field E₀ points along the a axis of the ellipse.

Here, let us first calculate the electrostatic potential Φ(ξ,η,ζ) of aconducting single-sheet hyperboloid with an elliptical hole, which canbe represented by a limiting hyperboloid from a family of hyperboloidsdescribed by the implicit equation

$\begin{matrix}{{\frac{x^{2}}{a^{2} + u} + \frac{y^{2}}{b^{2} + u} + \frac{z^{2}}{c^{2} + u}} = 1} & (133)\end{matrix}$

for a>b>c. The cubic roots ξ, η, and ζ are all real in the ranges

−a ² ≤ζ≤−b ² ,−b ² ≤η≤−c ² ,−c ²≤ξ<∞,  (134)

which are the ellipsoidal coordinates of a point (x, y, z). The limitinghyperboloid is a single planar sheet with an elliptical hole, i.e. itbelongs to the family of solutions η in the limit η→−c². Therefore, letus choose this limiting case as our origin in ellipsoidal coordinateswith c=0. Then Eq. (133) becomes

$\begin{matrix}{{\frac{x^{2}}{a^{2} + u} + \frac{y^{2}}{b^{2} + u} + \frac{z^{2}}{u}} = 1} & (135)\end{matrix}$

for a>b>c=0. The cubic roots ξ, η, and ζ are all real in the ranges

−a ² ≤ζ≤−b ² ,−b ²≤η≤0, 0≤ξ<∞,  (136)

The surface of the conducting hyperboloid is defined by −b²≤η=η₁<0. Letus consider the case E₀=E₀x, which in the limit when the hyperboloidbecomes a flat plane is the most relevant one. Therefore

$\begin{matrix}{\Psi_{0} = {{{- E_{0}}x} = {{\mp E_{0}}\sqrt{\frac{\left( {\xi + a^{2}} \right)\left( {\eta + a^{2}} \right)\left( {\zeta + a^{2}} \right)}{\left( {b^{2} - a^{2}} \right)\left( {- a^{2}} \right)}}}}} & (137)\end{matrix}$

in the lower-half plane, where the negative sign corresponds to positivex values and the positive sign to negative x values. Since theequipotential surfaces are determined by η, let Ψ_(p) be the potentialcaused by the hyperboloid, with the boundary condition Ψ_(in)(η=0)=0.Requiring continuous boundary condition on the surface of thehyperboloid, we have

$\begin{matrix}{{{\Psi_{in}\left( {\xi,\eta_{1},\zeta} \right)} = {{\Psi_{0}\left( {\xi,\eta_{1},\zeta} \right)} + {\Psi_{p}\left( {\xi,\eta_{1},\zeta} \right)}}},} & (138) \\{{{{ɛ_{in}\frac{\partial\Psi_{in}}{\partial\eta}}❘_{\eta_{1}}} = {{ɛ_{m}\frac{\partial\Psi_{0}}{\partial\eta}}❘_{\eta_{1}}{{{+ ɛ_{m}}\frac{\partial\Psi_{p}}{\partial\eta}}❘_{\eta_{1}}}}},} & (139)\end{matrix}$

where in the second equation the normal component of D at η=η₁ must becontinuous. Then we make the ansatz for the electrostatic potentialinside the hyperboloid,

Ψ_(in)(ξ,η,ζ)=−C _(in) E ₀ x,  (140)

where C_(in) is a constant. This ansatz satisfies the boundary conditionΨ_(in)(ξ=0,η→0,ζ)=0. For the outside polarization field we choose

Ψ_(p)(ξ,η,ζ)=−C _(p) E ₀ xF ₁(ξ)K ₁(η)  (141)

where C_(p) is a constant, and we defined

$\begin{matrix}\begin{matrix}{{F_{1}(\xi)} = {{\int\limits_{\xi}^{\infty}\frac{{ad}\;\xi^{\prime}}{2{\xi^{{\prime 1}\text{/}2}\left( {\xi^{\prime} + a^{2}} \right)}}} - {\int\limits_{\xi}^{\infty}\frac{{ad}\;\xi^{\prime}}{2\left( {\xi^{\prime} + a^{2}} \right)^{3\text{/}2}}}}} \\{= {{\arctan\left( \frac{a}{\sqrt{\xi}} \right)} - {\frac{a}{\sqrt{\xi + a^{2}}}.}}}\end{matrix} & (142)\end{matrix}$

Note that

${{\lim_{\xi\rightarrow 0_{+}}{\arctan\left( \frac{a}{\sqrt{\xi}} \right)}} = {\pi\text{/}2}},$

whereas

${\lim_{\xi\rightarrow 0_{-}}{\arctan\left( \frac{a}{\sqrt{\xi}} \right)}} = {{- \pi}\text{/}2.}$

Therefore, in order to avoid discontinuity at ξ=0, we must have

${\arctan\left( \frac{a}{- \sqrt{\xi}} \right)} = {\pi - {{\arctan\left( \frac{a}{\sqrt{\xi}} \right)}.}}$

$\begin{matrix}{{{K_{1}(\eta)} = {\int_{\eta}^{\infty}\frac{d\;\eta^{\prime}}{\left( {\eta^{\prime} + a^{2}} \right)R_{\eta^{\prime}}}}},} & (143)\end{matrix}$

where R_(η)=√{square root over ((η+a²)(η+b²)(−η))}. The boundaryconditions at z→±∞ are satisfied:

$\begin{matrix}{{F_{1}(\xi)} = \left\{ {\begin{matrix}0 & \left. {{for}\mspace{14mu} z}\rightarrow{+ \infty} \right. \\\pi & \left. {{for}\mspace{14mu} z}\rightarrow{- \infty} \right.\end{matrix}.} \right.} & (144)\end{matrix}$

At large distances r=√{square root over (x²+y²+z²)} from the wormhole wehave ξ≈r². Then the far-field potential in the upper half-space, whichis given by the pure polarization field, is

$\begin{matrix}{{\Psi_{p}\left( {\xi,\eta,\zeta} \right)} \approx {{- C_{p}}E_{0}{{xK}_{1}\left( {\eta = {- b^{2}}} \right)}\frac{1}{3}\left( \frac{a}{\sqrt{\xi}} \right)^{3}} \approx {{- C_{p}}E_{0}{K_{1}\left( {\eta \approx {- b^{2}}} \right)}\frac{a^{3}}{3}{\frac{x}{r^{3}}.}}} & (145)\end{matrix}$

The polarization far-field has the form of a dipole field at largedistances r from the wormhole. In order to determine the polarizabilityof the wormhole, let us find the solution at ξ=0, corresponding to theplane that passes through the center of the wormhole. For ξ=0, the unitvectors x and η are parallel. In this near-field limit, the polarizationpotential has the form

Ψ_(p)(ξ,η,ζ)=−{tilde over (C)} _(p) E ₀ xK ₁(η),  (146)

where {tilde over (C)}_(p)=C_(p)(π/2−1).

Using the boundary conditions shown in Eq. (139), we obtain the firstequation

{tilde over (C)} _(p) K ₁(η₁)−C _(in)=1,  (147)

and the second equation

$\begin{matrix}{{{{ɛ_{m}{{\overset{\sim}{C}}_{p}\left\lbrack {{{{K_{1}\left( \eta_{1} \right)}\frac{\partial x}{\partial\eta}}❘_{\eta_{1}}{+ K_{1}}},{{\left( \eta_{1} \right)x}❘_{\eta_{1}}}} \right\rbrack}} - {ɛ_{in}C_{in}\frac{\partial x}{\partial\eta}}}❘_{\eta_{1}}} = {{ɛ_{m}\frac{\partial x}{\partial\eta}}❘_{\eta_{1}}.}} & (148)\end{matrix}$

Using the derivatives

$\begin{matrix}{{{\frac{\partial x}{\partial\eta}❘_{{\xi = 0},\eta_{1}}} = {\frac{a}{2}\sqrt{\frac{\left( {\zeta + a^{2}} \right)}{\left( {\eta_{1} + a^{2}} \right)\left( {a^{2} - b^{2}} \right)\left( {a^{2} - c^{2}} \right)}}}},} & (149) \\{K_{1},{\left( \eta_{1} \right) = \frac{1}{\left( {\eta_{1} + a^{2}} \right)R_{\eta_{1}}}}} & (150)\end{matrix}$

we can rewrite the second equation as

$\begin{matrix}{{{{ɛ_{m}{{\overset{\sim}{C}}_{p}\left\lbrack {{\frac{K_{1}\left( \eta_{1} \right)}{\eta_{1} + a^{2}} + K_{1}},\left( \eta_{1} \right)} \right\rbrack}} - {ɛ_{in}C_{in}\frac{1}{\eta_{1} + a^{2}}}} = {ɛ_{m}\frac{1}{\eta_{1} + a^{2}}}},} & (151)\end{matrix}$

which is equivalent to

$\begin{matrix}{{{ɛ_{m}{{\overset{\sim}{C}}_{p}\left\lbrack {{K_{1}\left( \eta_{1} \right)} + \frac{1}{R_{\eta_{1}}}} \right\rbrack}} - {ɛ_{in}C_{in}}} = {ɛ_{m}.}} & (152)\end{matrix}$

Thus, the potentials are

$\begin{matrix}{{\Psi_{in} = \frac{\Psi_{0}}{1 + \frac{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}},} & (153) \\{\Psi_{p} = {\Psi_{0}{\frac{R_{\eta_{1}}\frac{ɛ_{m} - ɛ_{in}}{ɛ_{m}}{F_{1}(\xi)}{K_{1}(\eta)}\left( {{\pi\text{/}2} - 1} \right)}{1 + \frac{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}.}}} & (154)\end{matrix}$

Then the far-field potential in the upper half-space, which is given bythe pure polarization field, is

$\begin{matrix}\begin{matrix}{\Psi_{p} \approx {{- E_{0}}\frac{R_{\eta_{1}}\frac{ɛ_{m} - ɛ_{in}}{ɛ_{m}}{K_{1}\left( {\eta \approx {- b^{2}}} \right)}\left( {{\pi\text{/}2} - 1} \right)}{1 + \frac{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}\frac{a^{3}}{3}\frac{x}{r^{3}}}} \\{\approx {{- E_{0}}\frac{{ab}\sqrt{- \eta_{1}}\frac{ɛ_{m} - ɛ_{in}}{ɛ_{m}}\frac{2\pi}{a^{3}}\left( {{\pi\text{/}2} - 1} \right)}{1 + \frac{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}\frac{a^{3}}{3}\frac{x}{r^{3}}}} \\{{= {{- E_{0}}\frac{{ab}\sqrt{- \eta_{1}}\frac{ɛ_{m} - ɛ_{in}}{ɛ_{m}}{\pi\left( {{\pi\text{/}2} - 1} \right)}}{1 + \frac{L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}\frac{2x}{3r^{3}}}},}\end{matrix} & (155)\end{matrix}$

where we assumed that a≈b. The polarization far-field has the form of adipole field at large distances r from the wormhole. If the externalelectric field is applied in y-direction, we obtain the potentials

$\begin{matrix}{{\Psi_{in} = \frac{\Psi_{0}}{1 + \frac{L_{2}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}},} & (156) \\{{\Psi_{p} = {\Psi_{0}\frac{R_{\eta_{1}}\frac{ɛ_{m} - ɛ_{in}}{ɛ_{m}}{F_{2}(\xi)}{K_{2}(\eta)}\left( {\pi - 1} \right)}{1 + \frac{L_{2}\left( {ɛ_{in} - ɛ_{m}} \right)}{ɛ_{m}}}}},{with}} & (157) \\{{{F_{2}(\xi)} = {{\int_{\xi}^{\infty}\frac{{bd}\;\xi^{\prime}}{2^{\xi^{\prime}1\text{/}2}\left( {\xi^{\prime} + b^{2}} \right)}} - {\int_{\xi}^{\infty}\frac{{bd}\;\xi^{\prime}}{2\left( {\xi^{\prime} + b^{2}} \right)^{3\text{/}2}}}}},} & (158) \\{{K_{2}(\eta)} = {\int_{\eta}^{\infty}{\frac{d\;\eta^{\prime}}{\left( {\eta^{\prime} + b^{2}} \right)R_{\eta^{\prime}}}.}}} & (159)\end{matrix}$

We defined the geometrical factors

$\begin{matrix}{{L_{1} = {{R_{\eta_{1}}{K_{1}\left( \eta_{1} \right)}} \approx {{ab}\sqrt{- \eta_{1}}{\int_{\eta_{1}}^{\infty}\frac{d\;\eta^{\prime}}{\left( {\eta^{\prime} + a^{2}} \right)R_{\eta^{\prime}}}}}}},} & (160) \\{{L_{2} = {{R_{\eta_{1}}{K_{2}\left( \eta_{1} \right)}} \approx {{ab}\sqrt{- \eta_{1}}{\int_{\eta_{1}}^{\infty}\frac{d\;\eta^{\prime}}{\left( {\eta^{\prime} + b^{2}} \right)R_{\eta^{\prime}}}}}}},} & (161)\end{matrix}$

which are related to the depolarization factors by

$\begin{matrix}{{\overset{\sim}{L}}_{i} = {\frac{ɛ_{in} - ɛ_{m}}{ɛ_{in} - ɛ_{0}}{\frac{L_{i}}{ɛ_{m}}.}}} & (162)\end{matrix}$

This result determines the polarizability of the uncharged hyperboloidobservable in the far-field, i.e.

$\begin{matrix}{\alpha_{1} = {\frac{2{ab}\sqrt{- \eta_{1}}{\pi\left( {{\pi\text{/}2} - 1} \right)}}{3}{\frac{ɛ_{in} - ɛ_{m}}{ɛ_{m} + {L_{1}\left( {ɛ_{in} - ɛ_{m}} \right)}}.}}} & (163)\end{matrix}$

Similarly, we obtain the polarizability in y-direction, i.e.

$\begin{matrix}{\alpha_{2} = {\frac{2{ab}\sqrt{- \eta_{1}}{\pi\left( {{\pi\text{/}2} - 1} \right)}}{3}{\frac{ɛ_{in} - ɛ_{m}}{ɛ_{m} + {L_{2}\left( {ɛ_{in} - ɛ_{m}} \right)}}.}}} & (164)\end{matrix}$

Comparing to the polarizabilities of ellipsoids, [4] thepolarizabilities of hyperboloids are proportional to ab√{square rootover (−η₁)}, which corresponds to the volume of the ellipsoid abc. Inthe case of circular wormholes, we have a=b, and therefore α₁=α₂=α_(∥),with L₁=L₂=L_(∥).

In our proposed mid-IR light source the effective combination of siliconnitride (Si₃N₄) and hexagonal boron nitride (h-BN) behaves as anenvironment with polar phonons. Both materials are polar with ions ofdifferent valence, which leads to the Frohlich interaction betweenelectrons and optical phonons. [29] FIG. 17 shows that the interactionbetween the electrons in graphene and the polar substrate/graphenephonons modifies substantially the dispersion relations for the surfaceplasmon polaritons in graphene. The RPA dielectric function of grapheneis given by [24, 22]

$\begin{matrix}{{ɛ^{RPA}\left( {q,\omega} \right)} = {ɛ_{m} - {{v_{c}(q)}{\chi^{0}\left( {q,\omega} \right)}} - {ɛ_{m}{\sum\limits_{l}{{v_{{sph},l}\left( {q,\omega} \right)}{\chi^{0}\left( {q,\omega} \right)}}}} - {ɛ_{m}{v_{oph}\left( {q,\omega} \right)}{{\chi_{j,j}^{0}\left( {q,\omega} \right)}.}}}} & (165)\end{matrix}$

The second term is due to the effective Coulomb interaction, andv_(c)(q)=e²/2qε₀ is the 2D Coulomb interaction. The effectiveelectron-electron interaction mediated by the substrate optical phonons,

v _(sph,l)(q,ω)=|M _(sph)|² G _(l) ⁰(ω),  (166)

gives rise to the third term, where |M_(sph)|² is the scattering andG_(l) ⁰ is the free phonon Green function. The effectiveelectron-electron interaction due to the optical phonons in graphene,

v _(oph)(q,ω)=|M _(oph)|² G ⁰(ω),  (167)

gives rise to the last term of Eq. (165). |M_(o)ph|² is the scatteringmatrix element, and G^(o)(ω) is the free phonon Green function. χ_(j,j)⁰(q,ω) is the current-current correlation function in Eq. (165).

FIG. 18 includes a diagram 580 showing the energy loss function forgraphene with ϵ_(F)=−1.0 eV. k_(LSP4), k_(LSP7), and k_(LSP10) are theplasmon wavenumbers associated with the nanopatterning of the graphenesheet shown in FIGS. 5, 6, and 8, respectively. ω_(LSP4), ω_(LSP7), andω_(LSP10) represent the LSP resonances shown in FIGS. 5, 6, and 8,respectively. The polar phonon resonance of h-BN and the surface polarphonon resonance of Si₃N₄ are denoted by ω_(BN), and ω_(SN),respectively. The Landau damping region is marked by the shaded area.

The relaxation time τ of the momentum consists of the electron-impurity,electron-phonon, and the electron-edge scattering, τ⁻¹=τ_(DC)⁻¹+τ_(edge) ⁻¹+τ_(e-p) ⁻¹, which determines the plasmon lifetime and theabsorption spectrum bandwidth. It can be evaluated via the measured DCmobility μ of the graphene sample through τ_(DC)=μℏ√{square root over(πρ)}/ev_(F), where v_(F)=10⁶ m/s is the Fermi velocity, and the chargecarrier density is given by τ_(DC)=μℏ√{square root over (πρ)}/eσ_(F).The edge scattering time is τ_(edge)≈(1×10⁶(m/s)/w−w₀)⁻¹, where w is theedge-to-edge distance between the holes, and w₀=7 nm is the adjustmentparameter. The electron-phonon scattering time isτ_(e-ph)=ℏ/2Im(Σ_(e-ph)). The imaginary part of the electron-phononself-energy reads

Im(Σ_(e-ph))=γ|ℏω−sgn(ℏω−E _(F))ℏω_(oph)|,  (168)

where γ=18.3×10⁻³ is the electron-phonon coupling coefficient. Theoptical phonon energy of graphene is given by ℏω_(oph)≈0.2 eV.

The loss function Z describes the interaction of the SPPs and thesubstrate/graphene phonons. In RPA we have

$\begin{matrix}{Z \propto {- {{{Im}\left( \frac{1}{ɛ^{RPA}} \right)}.}}} & (169)\end{matrix}$

In FIG. 17 (diagram 575), line a shows the loss function for graphenewith carrier mobility μ=3000 cm²/V·s and a Fermi energy of ϵ_(F)=1.0 eV.In order to take advantage of the enhancement of the electromagneticfield at the position of the graphene sheet, the thickness of theoptical cavity must be λ/4n, where n is the refractive index of thecavity material. [24] The LSP resonance frequencies ω_(LSP4), ω_(LSP7),and ω_(LSP10) mark the frequencies around the resonance wavelengths of 4μm, 7 μm. and 10 μm. The resonance frequencies of the polar phonons aredenoted by ω_(BN) for h-BN and by ω_(SN) for Si₃N₄.

For the calculation of the spectral radiance we need to integrate theelements of the dyadic Green function over the spherical angle. We cansplit the total dyadic Green function into a free space term

₀(r,r′;ω) and a term

_(SPP)(r,r′;ω) that creates surface plasmon polaritons inside graphene.Since the absorbance of the pristine graphene sheet is only 2.3%, we cansafely neglect

_(SPP)(r,r′;ω). Our goal is to calculate the gray-body emission of theEM radiation from the LSP around the holes in graphene into free space.Therefore, we need to evaluate

$\begin{matrix}{{{I_{GB}^{\infty}(\omega)} = {\lim\limits_{r\rightarrow\infty}{\int{r^{2}\sin\;\theta\; d\;\theta\; d\;\varphi\;{I_{GB}\left( {r,\omega} \right)}}}}},} & (170)\end{matrix}$

where can use the approximation

I _(GB)(r,ω)=I ₀(r,ω)−I _(SPP)(r,ω)≈I ₀(r,ω).  (171)

In Cartesian coordinates, we can write down the dyadic Green function as[21]

$\begin{matrix}{{{\overset{\leftrightarrow}{G}}_{0}\left( {r;\omega} \right)} = {{\frac{e^{ikr}}{4\pi\; r}\left\lbrack {{\left( {1 + \frac{i}{kr} - \frac{1}{k^{2}r^{2}}} \right)\overset{\leftrightarrow}{11}} + {\left( {\frac{3}{k^{2}r^{2}} - \frac{3i}{kr} - 1} \right){rr}}} \right\rbrack}.}} & (172)\end{matrix}$

Since we are interested only in the far field, we consider only thefar-field component of the dyadic Green function, which is

$\begin{matrix}{{{{\overset{\leftrightarrow}{G}}_{FF}\left( {r;\omega} \right)} = {\frac{e^{ikr}}{4\pi\; r}\left\lbrack {\overset{\leftrightarrow}{11} - {rr}} \right\rbrack}},} & (173)\end{matrix}$

which possesses only angular (transverse) components but no radial(longitudinal) components. Then the necessary components are

$\begin{matrix}{{{G_{xx}\left( {r;\omega} \right)} = {\frac{e^{ikr}}{4\pi\; r}\left\lbrack {1 - {\sin^{2}\theta\;\cos^{2}\varphi}} \right\rbrack}},{{G_{yx}\left( {r;\omega} \right)} = {\frac{e^{ikr}}{4\pi\; r}\left\lbrack {1 - {\sin^{2}\theta\;\cos\;\varphi\;\sin\;\varphi}} \right\rbrack}},{{G_{zx}\left( {r;\omega} \right)} = {\frac{e^{ikr}}{4\pi\; r}\left\lbrack {1 - {\sin\;\theta\;\cos\;{\theta cos}\;\varphi}} \right\rbrack}},{{G_{xy}\left( {r;\omega} \right)} = {\frac{e^{ikr}}{4\pi\; r}\left\lbrack {1 - {\sin^{2}\theta\;\cos\;{\varphi sin}\;\varphi}} \right\rbrack}},{{G_{yy}\left( {r;\omega} \right)} = {\frac{e^{ikr}}{4\pi\; r}\left\lbrack {1 - {\sin^{2}\theta\;\sin^{2}\varphi}} \right\rbrack}},{{G_{zy}\left( {r;\omega} \right)} = {\frac{e^{ikr}}{4\pi\; r}\left\lbrack {1 - {\sin\;\theta\;\cos\;{\theta sin}\;\varphi}} \right\rbrack}},} & (174)\end{matrix}$

The corresponding integrals are

$\begin{matrix}{{{\int{r^{2}\sin\;\theta\; d\;\theta\; d\;\varphi{{G_{xx}\left( {r;\omega} \right)}}^{2}}} = \frac{2}{15\pi}},{{\int{r^{2}\sin\;\theta\; d\;\theta\; d\;\varphi{{G_{yx}\left( {r;\omega} \right)}}^{2}}} = \frac{4}{15\pi}},{{\int{r^{2}\sin\;\theta\; d\;\theta\; d\;\varphi{{G_{zx}\left( {r;\omega} \right)}}^{2}}} = \frac{4}{15\pi}},{{\int{r^{2}\sin\;\theta\; d\;\theta\; d\;\varphi{{G_{xy}\left( {r;\omega} \right)}}^{2}}} = \frac{4}{15\pi}},{{\int{r^{2}\sin\;\theta\; d\;\theta\; d\;\varphi{{G_{yy}\left( {r;\omega} \right)}}^{2}}} = \frac{2}{15\pi}},{{\int{r^{2}\sin\;\theta\; d\;\theta\; d\;\varphi{{G_{zy}\left( {r;\omega} \right)}}^{2}}} = {\frac{4}{15\pi}.}}} & (175)\end{matrix}$

Regarding the doping of graphene due to Si₃N₄, the Silicon nitride,Si₃N₄, dielectric layer causes an effective n-type doping in graphenesheet [34, 35]. The shift in Fermi energy is given by

E _(F) =

v _(F) √{square root over (πn)},  (176)

where vF is the Fermi velocity (vF≈10⁶ m/s for graphene), is Planck'sconstant, and n is the carrier density. The carrier density n depends onthe gate voltage and capacitance, i.e. where V=(V_(G)−V_(CNP)) is thegate voltage relative to charge neutrality point, e is electric charge,and C is the capacitance of dielectric layer, given by C=ε_(r)ε₀/d,ε_(r) is the relative permittivity, ε₀ is the permittivity of freespace, and d is the thickness of dielectric layer.

The gate capacitance for a 50 nm thick Si₃N₄ layer in the infraredregion is VG=4.5×10⁻⁸ F/cm². From Eq. (176) we conclude that the Fermienergy E_(F)=1 eV corresponds to a gate voltage relative to the CNP ofV=(V_(G)−V_(CNP))=6.9 V. Wang et al. [36] observed that a Si₃N₄ filmwith a thickness of 50 nm shifts the CNP in a graphene sheet to −20 V,which shows that graphene is n-doped at zero gate voltage and the Fermienergy is E_(F)=1.74 eV. The Fermi energy can be tuned by applying agate voltage to a desired value. In our work, we have used a Fermienergy of E_(F)=1 eV, which corresponds to V=6.59 V, i.e. for the CNP at−20 V, V_(G)=−13.41 V results in a Fermi energy of E_(F)=1 eV. From Eqs.(176) and (177), the carrier density required to achieve a Fermi energyof E_(F)=1 eV is n=1.94×1012 cm⁻², which corresponds to an electricfield of E_(1.0 eV)=en/ε_(r)ε₀=1.38×10⁶ Vcm⁻¹, which is in the safe zonecompared to the reported breakdown field of the order of 10⁷ Vcm⁻¹ [38].

Referring now additionally to FIGS. 19A-19B, an IR source 100 accordingto the present disclosure is now described. The IR source 100illustratively includes an electrically conductive layer 101 defining aback contact. The electrically conductive layer 101 may comprise atleast one of gold, silver, and platinum. The electrically conductivelayer 101 is coupled to a reference voltage, for example, theillustrated ground potential.

The IR source 100 illustratively comprises a first dielectric layer 102over the electrically conductive layer 101, a transparent electricallyconductive layer 103 over the first dielectric layer, and a seconddielectric layer 104 over the transparent electrically conductive layer.The first dielectric layer 102 may comprise a polymer dielectricmaterial (e.g. SU-8 photoresist), but may comprise any dielectricmaterial that is transparent to mid-IR. The first dielectric layer 102defines a Fabry-Perot cavity, and must be transparent in the operationalfrequency of the IR source 100, for example, mid-IR. The transparentelectrically conductive layer 103 may comprise indium tin oxide (ITO),or any conductive material transparent to mid-IR. Indeed, thetransparent electrically conductive layer 103 must be transparent in theoperational frequency of the IR source 100, for example, mid-IR. Thesecond dielectric layer 104 may comprise an oxide layer, for example,silicon dioxide, or a silicon nitride material.

The IR source 100 illustratively includes a graphene layer 105 over thesecond dielectric layer 104 and having a perforated pattern 106. Theperforated pattern 106 may comprise an array of circular-openings (SeeFIGS. 1-2). In the illustrated embodiment, the perforated pattern 106comprises an array of elliptical holes 107 a-107 p. Also, the array isillustratively square-shaped, but can take on other shapes, such as arectangle or a curved boundary shape.

The IR source 100 illustratively includes a protective layer 110 overthe graphene layer 105, and first and second electrically conductivecontacts 111 a-111 b coupled to the graphene layer. The protective layer110 may comprise boron-nitride, for example, and protects the graphenelayer 105 from oxidation.

The IR source 100 illustratively includes a voltage source 112configured to apply a voltage signal is applied between the first andsecond electrically conductive contacts 111 a-111 b. The graphene layer105 is configured to emit IR radiation 113 in a frequency range when thevoltage signal is applied between the first and second electricallyconductive contacts 111 a-111 b (e.g. copper, aluminum, silver, gold, orplatinum). Also, the graphene layer 105 may be configured to emit the IRradiation 113 at an angle α from normal based upon varying the voltagesignal. For example, the graphene layer 105 may be configured to emitmid-IR radiation. The graphene layer 105 may be configured to emit theIR radiation 113 at the angle within a range of 12°-80° from normalbased upon the voltage signal. The graphene layer 105 may be configuredto selectively change the frequency range based upon the voltage signal.

The graphene layer 105 is configured to emit IR radiation that ispartially coherent (See FIG. 11). The graphene layer 105 is configuredto emit the IR radiation 113 at a selective wavelength, which is basedupon the size of the openings in the perforated pattern 106 in thegraphene layer 105. The gate voltage may also be applied selectively toadjust the frequency of the IR radiation 113. In an example embodiment,the size of the IR source 100 may comprise 10 μm×10 μm.

Referring now additionally to FIG. 20, another embodiment of the IRsource 100 configured as a phased array 200 is now described. In thisembodiment of the phased array 200, those elements already discussedabove with respect to FIGS. 19A-19B are incremented by 100 and mostrequire no further discussion herein. This embodiment differs from theprevious embodiment in that this phased array 200 illustrativelyincludes an electrically conductive layer 201, a first dielectric layer202 over the electrically conductive layer, and a transparentelectrically conductive layer 203 over the first dielectric layer. Thephased array also comprises a second dielectric layer 204 over thetransparent electrically conductive layer 203, and a graphene layer 205over the second dielectric layer and having a perforated pattern 206comprising an array of elliptical holes. The graphene layer 205 isconfigured to emit IR radiation in a frequency range.

The phased array also comprises first and second electrically conductivecontacts 211 a-211 b coupled to the graphene layer 205, and circuity212. The circuity 212 is configured to apply a voltage signal betweenthe first and second electrically conductive contacts 211 a-211 b,change the voltage signal to selectively set phase change φ(a)-φ(e)between the IR radiation 213 emitted from adjacent elliptical holes 207a-207 e to emit the IR radiation at an angle from normal, and change thevoltage signal to selectively set the frequency range of the IRradiation.

Referring now additionally to FIG. 21, a method for making an IR source100 is now described with reference to a flowchart 1000, which begins atBlock 1001. The method comprises forming a first dielectric layer 102over an electrically conductive layer 101 (Block 1003), forming atransparent electrically conductive layer 103 over the first dielectriclayer (Block 1005), and forming a second dielectric layer 104 over thetransparent electrically conductive layer (Block 1007). The methodcomprises forming a graphene layer 105 over the second dielectric layer104 and having a perforated pattern (Block 1009), forming a protectivelayer 110 over the graphene layer, and forming first and secondelectrically conductive contacts 111 a-111 b coupled to the graphenelayer (Block 1011). The graphene layer 105 is configured to emit IRradiation in a frequency range when a voltage signal is applied betweenthe first and second electrically conductive contacts 111 a-111 b. Themethod ends at Block 1013.

Referring now to FIGS. 20 and 22, a method for operating a phased array200 is now described with reference to a flowchart 2000, which begins atBlock 2001. The phased array comprises 200 an electrically conductivelayer 201, a first dielectric layer 202 over the electrically conductivelayer, a transparent electrically conductive layer 203 over the firstdielectric layer, a second dielectric layer 204 over the transparentelectrically conductive layer, a graphene layer 205 over the seconddielectric layer and having a perforated pattern 206 comprising an arrayof elliptical holes 207 a-207 e, the graphene layer configured to emitIR radiation 213 in a frequency range, and first and second electricallyconductive contacts 2111-211 b coupled to the graphene layer. The methodincludes operating circuity 212 coupled between the first and secondelectrically conductive contacts 211 a-211 b to apply a voltage signalbetween the first and second electrically conductive contacts (Block2003), change the voltage signal to selectively set phase change betweenthe IR radiation 213 emitted from adjacent elliptical holes 207 a-207 eto emit the IR radiation at an angle from normal (Block 2005), andchange the voltage signal to selectively set the frequency range of theIR radiation (Block 2007). The method ends at Block 2009.

Many modifications and other embodiments of the present disclosure willcome to the mind of one skilled in the art having the benefit of theteachings presented in the foregoing descriptions and the associateddrawings. Therefore, it is understood that the present disclosure is notto be limited to the specific embodiments disclosed, and thatmodifications and embodiments are intended to be included within thescope of the appended claims.

REFERENCES (INCORPORATED BY REFERENCE IN THEIR ENTIRETY)

[1] Milton Abramowitz and Irene A. Stegun. Handbook of MathematicalFunctions: with Formulas, Graphes, and Mathematical Tables. Dover, 1965.

[2] Bandurin, D. A. et al. Negative local resistance caused by viscouselectron backflow in graphene. Science 351, 1055-1058 (2016).

[3] Baranov, Denis G. et al. Nanophotonic engineering of far-fieldthermal emitters. Nat. Mater. 18, 920-930 (2019).

[4] Bohren, C. F. & Huffman, D. R. Absorption and scattering of light bysmall particles (Wiley, Amsterdam, 1998).

[5] Rémi Carminati and Jean-Jacques Greffet. Near-field effects inspatial coherence of thermal sources. Phys. Rev. Lett., 82:1660-1663,February 1999.

[6] Celanovic, Ivan, Perreault, David & Kassakian, John. Resonant-cavityenhanced thermal emission. Phys. Rev. B 72, 075127 (2005).

[7] Christensen, Thomas. From Classical to Quantum Plasmonics in Threeand Two Dimensions (Springer, Berlin, 2017).

[8] Cornelius, Christopher M. & Dowling, Jonathan P. Modification ofplanck blackbody radiation by photonic band-gap structures. Phys. Rev. A59, 4736-4746 (1999).

[9] Freitag, M., Chiu, H. Y., Steiner, M., Perebeinos, V. & Avouris, P.Thermal infrared emission from biased graphene. Nat. Nanotechnol. 5, 497(2010).

[10] Greffet, Jean-Jacques. et al. Coherent emission of light by thermalsources. Nature 416, 61-64 (2002).

[11] Greffet, Jean-Jacques. Revisiting thermal radiation in the nearfield. C.R. Phys. 18, 24-30 (2017).

[12] Carminati, Rémi. & Greffet, Jean-Jacques. Near-field effects inspatial coherence of thermal sources. Phys. Rev. Lett. 82, 1660-1663(1999).

[13] Henkel, C. & Joulain, K. Electromagnetic field correlations near asurface with a nonlocal optical response. Appl. Phys. B 84, 61-68(2006).

[14] Kim, Y. D. et al. Ultrafast graphene light emitters. Nano Lett. 18,934-940 (2018).

[15] Landau, L. D., Lifshitz, E. M. & Pitaevskii, L. P. Electrodynamicsof Continuous Media 2nd edn. (Elsevier, Amsterdam, 1984).

[16] Lifshitz, Evgeny & Pitaevskii, P. Statistical Physics: Part 2 3rdedn. (Elsevier, Amsterdam, 1980).

[17] Lin, Shawn-Yu. et al. Enhancement and suppression of thermalemission by a three-dimensional photonic crystal. Phys. Rev. B 62,R2243-R2246 (2000).

[18] Li, Le-Wei., Kang, Xiao-Kang. & Leong, Mook-Seng. Spheroidal WaveFunctions in Electromagnetic Theory (Wiley, Hoboken, 2002).

[19] Yang, Zih-Ying. et al. Narrowband wavelength selective thermalemitters by confined tamm plasmon polaritons. ACS Photonics 4, 2212-2219(2017).

[20] Luxmoore, I. J. et al. Thermal emission from large area chemicalvapor deposited graphene devices. Appl. Phys. Lett. 103, 131906 (2013).

[21] Novotny, Lukas & Hecht, Bert. Principles of Nano-Optics (CambridgeUniversity Press, Cambridge, 2012).

[22] Paudel, Hari P., Safaei, Alireza & Michael, N. Leuenberger,nanoplasmonics in metallic nanostructures and dirac systems. InNanoplasmonics—Fundamentals and Applications 3rd edn (ed. Grégory, B.)1142 (Intech, London, 2017).

[23] Roy, Supti, Perez-Guaita, David, Bowden, Scott, Heraud, Philip &Wood, Bayden R. Spectroscopy goes viral: Diagnosis of hepatitis b and cvirus infection from human sera using atr-ftir spectroscopy. Clin.Spectrosc. 1, 100001 (2019).

[24] Safaei, Alireza et al. Dynamically tunable extraordinary lightabsorption in monolayer graphene. Phys. Rev. B 96, 165431 (2017).

[25] A. Safaei, S. Chandra, M. W. Shabbir, M. N. Leuenberger, and D.Chanda. Dirac plasmon-assisted asymmetric hot carrier generation forroom-temperature infrared detection. Nature Communications, 10:3498,2019.

[26] Shiue, R. J. et al. Thermal radiation control from hot grapheneelectrons coupled to a photonic crystal nanocavity. Nat. Commun. 10, 109(2019).

[27] Thongrattanasiri, S., Koppens, F. H. L. & de Abajo, F. J. G.Complete optical absorption in periodically patterned graphene. Phys.Rev. Lett. 108, 047401 (2012).

[28] Zih-Ying Yang, Satoshi Ishii, Takahiro Yokoyama, Thang Duy Dao,Mao-Guo Sun, Pavel S. Pankin, Ivan V. Timofeev, Tadaaki Nagao, andKuo-Ping Chen. Narrowband wavelength selective thermal emitters byconfined tamm plasmon polaritons. ACS Photonics, 4(9):2212-2219, 2017.

[29] Yan, Hugen et al. Damping pathways of mid-infrared plasmons ingraphene nanostructures. Nat. Photonics 7, 394-399 (2013).

[30] Luo, Fang et al. Graphene thermal emitter with enhanced jouleheating and localized light emission in air. ACS Photonics 6, 2117-2125(2019).

[31] Zhao, L. L., Kelly, K. L. & Schatz, G. C. The extinction spectra ofsilver nanoparticle arrays: influence of array structure on plasmonresonance wavelength and width. J. Phys. Chem. B 107, 7343-7350 (2003).

[32] Zhang, Yuping et al. Independently tunable dual-band perfectabsorber based on graphene at mid-infrared frequencies, Sci. Rep. 5,18463 (2015).

[33] Lin, Keng-Te., Lin, Han, Yang, Tieshan & Jia, Baohua. Structuredgraphene metamaterial selective absorbers for high efficiency andomnidirectional solar thermal energy conversion. Nat. Commun. 11, 1389(2020).

[34] Henkel, C., Joulain, K., Carminati, R. & Greffet, J.-J. Spatialcoherence of thermal near fields. Opt. Commun. 186, 57-67 (2000).

[35] Safaei, A., Chandra, S., Leuenberger, M. N. & Chanda, D. Wide angledynamically tunable enhanced infrared absorption on large areananopatterned graphene. ACS Nano 13, 421-428 (2019).

[36] Wang, Zegao et al. Air-stable n-type doping of graphene fromoverlying si3n4 film. Appl. Surf. Sci. 307, 712-715 (2014).

[37] Wittmann, Sebastian et al. Dielectric surface charge engineeringfor electrostatic doping of graphene. ACS Appl. Electr. Mat. 2,1235-1242 (2020).

[38] Zhu, Wenjuan, Neumayer, Deborah, Perebeinos, Vasili & Avouris,Phaedon. Silicon nitride gate dielectrics and band gap engineering ingraphene layers. Nano Lett. 10, 3572-3576 (2010).

1. An infrared (IR) source comprising: an electrically conductive layer;a first dielectric layer over the electrically conductive layer; atransparent electrically conductive layer over the first dielectriclayer; a second dielectric layer over the transparent electricallyconductive layer; a graphene layer over the second dielectric layer andhaving a perforated pattern; and first and second electricallyconductive contacts coupled to the graphene layer; the graphene layerconfigured to emit IR radiation in a frequency range when a voltagesignal is applied between the first and second electrically conductivecontacts.
 2. The IR source of claim 1 wherein the graphene layer isconfigured to emit the IR radiation at an angle from normal based uponthe voltage signal.
 3. The IR source of claim 2 wherein the graphenelayer is configured to emit the IR radiation at the angle within a rangeof 12°-80° from normal based upon the voltage signal.
 4. The IR sourceof claim 1 wherein the graphene layer is configured to selectivelychange the frequency range based upon the voltage signal.
 5. The IRsource of claim 1 wherein the perforated pattern comprises an array ofelliptical holes.
 6. The IR source of claim 1 wherein the graphene layeris configured to emit mid-IR radiation.
 7. The IR source of claim 1wherein the electrically conductive layer comprise at least one of gold,silver, and platinum.
 8. The IR source of claim 1 wherein the firstdielectric layer comprises a polymer layer; and wherein the seconddielectric layer comprises an oxide layer.
 9. The IR source of claim 1wherein the electrically conductive layer is coupled to a referencevoltage.
 10. A phased array comprising: an electrically conductivelayer; a first dielectric layer over the electrically conductive layer;a transparent electrically conductive layer over the first dielectriclayer; a second dielectric layer over the transparent electricallyconductive layer; a graphene layer over the second dielectric layer andhaving a perforated pattern comprising an array of elliptical holes, thegraphene layer configured to emit infrared (IR) radiation in a frequencyrange; first and second electrically conductive contacts coupled to thegraphene layer; and circuity configured to apply a voltage signalbetween the first and second electrically conductive contacts, changethe voltage signal to selectively set phase change between the IRradiation emitted from adjacent elliptical holes to emit the IRradiation at an angle from normal, and change the voltage signal toselectively set the frequency range of the IR radiation.
 11. The phasedarray of claim 10 wherein the graphene layer is configured to emit theIR radiation at the angle within a range of 12°-80° from normal basedupon the voltage signal.
 12. The phased array of claim 10 wherein thegraphene layer is configured to emit mid-IR radiation.
 13. The phasedarray of claim 10 wherein the electrically conductive layer comprise atleast one of gold, silver, and platinum.
 14. The phased array of claim10 wherein the first dielectric layer comprises a polymer layer; andwherein the second dielectric layer comprises an oxide layer.
 15. Thephased array of claim 10 wherein the electrically conductive layer iscoupled to a reference voltage.
 16. A method for making an infrared (IR)source comprising: forming a first dielectric layer over an electricallyconductive layer; forming a transparent electrically conductive layerover the first dielectric layer; forming a second dielectric layer overthe transparent electrically conductive layer; forming a graphene layerover the second dielectric layer and having a perforated pattern; andforming first and second electrically conductive contacts coupled to thegraphene layer; the graphene layer configured to emit IR radiation in afrequency range when a voltage signal is applied between the first andsecond electrically conductive contacts.
 17. The method of claim 16wherein the graphene layer is configured to emit the IR radiation at anangle from normal based upon the voltage signal.
 18. The method of claim17 wherein the graphene layer is configured to emit the IR radiation atthe angle within a range of 12°-80° from normal based upon the voltagesignal.
 19. The method of claim 16 wherein the graphene layer isconfigured to selectively change the frequency range based upon thevoltage signal.
 20. The method of claim 16 wherein the perforatedpattern comprises an array of elliptical holes.
 21. A method ofoperating a phased array comprising an electrically conductive layer, afirst dielectric layer over the electrically conductive layer, atransparent electrically conductive layer over the first dielectriclayer, a second dielectric layer over the transparent electricallyconductive layer, a graphene layer over the second dielectric layer andhaving a perforated pattern comprising an array of elliptical holes, thegraphene layer configured to emit infrared (IR) radiation in a frequencyrange, and first and second electrically conductive contacts coupled tothe graphene layer, the method comprising: operating circuity coupledbetween the first and second electrically conductive contacts to apply avoltage signal between the first and second electrically conductivecontacts, change the voltage signal to selectively set phase changebetween the IR radiation emitted from adjacent elliptical holes to emitthe IR radiation at an angle from normal, and change the voltage signalto selectively set the frequency range of the IR radiation.
 22. Themethod of claim 21 further comprising operating circuity to cause thegraphene layer to emit the IR radiation at the angle within a range of12°-80° from normal based upon the voltage signal.
 23. The method ofclaim 21 further comprising operating circuity to cause the graphenelayer to emit mid-IR radiation.
 24. The method of claim 21 wherein theelectrically conductive layer comprise at least one of gold, silver, andplatinum.
 25. The method of claim 21 wherein the first dielectric layercomprises a polymer layer; and wherein the second dielectric layercomprises an oxide layer.